| Label |
Class |
Class size |
Class degree |
Base field |
Field degree |
Field signature |
Conductor |
Conductor norm |
Discriminant norm |
Root analytic conductor |
Bad primes |
Rank |
Torsion |
CM |
CM |
Sato-Tate |
$\Q$-curve |
Base change |
Semistable |
Potentially good |
Nonmax $\ell$ |
mod-$\ell$ images |
$Ш_{\textrm{an}}$ |
Tamagawa |
Regulator |
Period |
Leading coeff |
j-invariant |
Weierstrass coefficients |
Weierstrass equation |
| 19.1-a1 |
19.1-a |
$2$ |
$2$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{4} \) |
$4.02677$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1.617859637$ |
$28.25472648$ |
2.485597930 |
\( \frac{14554844437095}{130321} a^{2} - \frac{42094599319755}{130321} a - \frac{7684891448643}{130321} \) |
\( \bigl[a^{2} - a - 3\) , \( -a + 1\) , \( 0\) , \( 4 a - 12\) , \( -3 a^{2} + 21 a - 36\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(4a-12\right){x}-3a^{2}+21a-36$ |
| 19.1-a2 |
19.1-a |
$2$ |
$2$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$4.02677$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.808929818$ |
$56.50945296$ |
2.485597930 |
\( -\frac{679428}{361} a^{2} + \frac{4920021}{361} a + \frac{868752}{361} \) |
\( \bigl[a^{2} - a - 3\) , \( -a + 1\) , \( 0\) , \( -a + 3\) , \( 0\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(-a+3\right){x}$ |
| 19.1-b1 |
19.1-b |
$2$ |
$2$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$4.02677$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.133101439$ |
$328.0154753$ |
2.373974317 |
\( -\frac{679428}{361} a^{2} + \frac{4920021}{361} a + \frac{868752}{361} \) |
\( \bigl[a\) , \( -a^{2} + 3\) , \( a^{2} - a - 4\) , \( -a^{2} + 2 a + 3\) , \( a^{2} - 2 a - 3\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-a^{2}+2a+3\right){x}+a^{2}-2a-3$ |
| 19.1-b2 |
19.1-b |
$2$ |
$2$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
19.1 |
\( 19 \) |
\( - 19^{4} \) |
$4.02677$ |
$(a+3)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$0.266202878$ |
$164.0077376$ |
2.373974317 |
\( \frac{14554844437095}{130321} a^{2} - \frac{42094599319755}{130321} a - \frac{7684891448643}{130321} \) |
\( \bigl[a\) , \( -a^{2} + 3\) , \( a^{2} - a - 4\) , \( -11 a^{2} + 32 a + 8\) , \( 40 a^{2} - 115 a - 24\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-11a^{2}+32a+8\right){x}+40a^{2}-115a-24$ |
| 19.3-a1 |
19.3-a |
$2$ |
$3$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
19.3 |
\( 19 \) |
\( 19 \) |
$4.02677$ |
$(a^2-3a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$0.582416001$ |
$51.14851036$ |
3.239628527 |
\( \frac{33660}{19} a^{2} - \frac{206681}{19} a - \frac{34011}{19} \) |
\( \bigl[a^{2} - 4\) , \( -1\) , \( a + 1\) , \( -a^{2} + 2 a + 5\) , \( a + 1\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-a^{2}+2a+5\right){x}+a+1$ |
| 19.3-a2 |
19.3-a |
$2$ |
$3$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
19.3 |
\( 19 \) |
\( 19^{3} \) |
$4.02677$ |
$(a^2-3a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B |
$1$ |
\( 1 \) |
$1.747248004$ |
$17.04950345$ |
3.239628527 |
\( -\frac{3036479600000765}{6859} a^{2} - \frac{6268613843868843}{6859} a - \frac{990877707458156}{6859} \) |
\( \bigl[a^{2} - 4\) , \( -1\) , \( a + 1\) , \( -6 a^{2} + 7 a + 5\) , \( -41 a - 6\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a+1\right){y}={x}^{3}-{x}^{2}+\left(-6a^{2}+7a+5\right){x}-41a-6$ |
| 19.3-b1 |
19.3-b |
$2$ |
$3$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
19.3 |
\( 19 \) |
\( 19^{3} \) |
$4.02677$ |
$(a^2-3a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 3 \) |
$1$ |
$4.453723006$ |
0.484341998 |
\( -\frac{3036479600000765}{6859} a^{2} - \frac{6268613843868843}{6859} a - \frac{990877707458156}{6859} \) |
\( \bigl[a + 1\) , \( -a\) , \( a^{2} - a - 4\) , \( 4 a^{2} - 5 a - 30\) , \( 41 a^{2} - 46 a - 249\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}-a{x}^{2}+\left(4a^{2}-5a-30\right){x}+41a^{2}-46a-249$ |
| 19.3-b2 |
19.3-b |
$2$ |
$3$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
19.3 |
\( 19 \) |
\( 19 \) |
$4.02677$ |
$(a^2-3a-2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 1 \) |
$1$ |
$120.2505211$ |
0.484341998 |
\( \frac{33660}{19} a^{2} - \frac{206681}{19} a - \frac{34011}{19} \) |
\( \bigl[a + 1\) , \( -a\) , \( a^{2} - a - 4\) , \( -a^{2} + 5\) , \( -a^{2} + a + 5\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}-a{x}^{2}+\left(-a^{2}+5\right){x}-a^{2}+a+5$ |
| 21.1-a1 |
21.1-a |
$6$ |
$8$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{2} \cdot 7^{4} \) |
$4.09450$ |
$(-a-1), (a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$5.462546094$ |
$14.84310331$ |
2.204391663 |
\( -\frac{6753481671027081357094325833}{21609} a^{2} + \frac{1131188974087934247922313456}{3087} a + \frac{39155136968953218755956232668}{21609} \) |
\( \bigl[a + 1\) , \( -a^{2} + 3\) , \( a^{2} - 4\) , \( 8120 a^{2} - 9522 a - 47094\) , \( -656376 a^{2} + 769575 a + 3805513\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(8120a^{2}-9522a-47094\right){x}-656376a^{2}+769575a+3805513$ |
| 21.1-a2 |
21.1-a |
$6$ |
$8$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{16} \) |
$4.09450$ |
$(-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$5.462546094$ |
$3.710775828$ |
2.204391663 |
\( \frac{5050883273717386823929705}{299096375126409} a^{2} + \frac{1489574000087096330787088}{42728053589487} a + \frac{1647849108085064735158004}{299096375126409} \) |
\( \bigl[a + 1\) , \( -a^{2} + 3\) , \( a^{2} - 4\) , \( 490 a^{2} - 632 a - 2944\) , \( -9956 a^{2} + 11369 a + 57143\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(490a^{2}-632a-2944\right){x}-9956a^{2}+11369a+57143$ |
| 21.1-a3 |
21.1-a |
$6$ |
$8$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{8} \) |
$4.09450$ |
$(-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$2.731273047$ |
$29.68620662$ |
2.204391663 |
\( -\frac{110650233343082241553}{466948881} a^{2} + \frac{18533606397598137617}{66706983} a + \frac{641524754970175127041}{466948881} \) |
\( \bigl[a + 1\) , \( -a^{2} + 3\) , \( a^{2} - 4\) , \( 505 a^{2} - 597 a - 2939\) , \( -9858 a^{2} + 11554 a + 57146\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(505a^{2}-597a-2939\right){x}-9858a^{2}+11554a+57146$ |
| 21.1-a4 |
21.1-a |
$6$ |
$8$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{16} \cdot 7^{2} \) |
$4.09450$ |
$(-a-1), (a-1)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$2.731273047$ |
$7.421551657$ |
2.204391663 |
\( \frac{1063395459790548094577}{2109289329} a^{2} - \frac{439327317478130560369}{301327047} a - \frac{562061870493043570577}{2109289329} \) |
\( \bigl[a + 1\) , \( -a^{2} + 3\) , \( a^{2} - 4\) , \( 35 a^{2} - 37 a - 219\) , \( -48 a^{2} + 62 a + 248\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(35a^{2}-37a-219\right){x}-48a^{2}+62a+248$ |
| 21.1-a5 |
21.1-a |
$6$ |
$8$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$4.09450$ |
$(-a-1), (a-1)$ |
$1$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1.365636523$ |
$59.37241325$ |
2.204391663 |
\( \frac{582023899160}{15752961} a^{2} - \frac{1038165632761}{2250423} a + \frac{16832160366340}{15752961} \) |
\( \bigl[a + 1\) , \( -a^{2} + 3\) , \( a^{2} - 4\) , \( 30 a^{2} - 37 a - 179\) , \( -129 a^{2} + 150 a + 745\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(30a^{2}-37a-179\right){x}-129a^{2}+150a+745$ |
| 21.1-a6 |
21.1-a |
$6$ |
$8$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{2} \) |
$4.09450$ |
$(-a-1), (a-1)$ |
$1$ |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$0.682818261$ |
$118.7448265$ |
2.204391663 |
\( \frac{40304401}{3969} a^{2} + \frac{11995048}{567} a + \frac{11325248}{3969} \) |
\( \bigl[a + 1\) , \( -a^{2} + 3\) , \( a^{2} - 4\) , \( -2 a - 4\) , \( -2 a^{2} + a + 9\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+3\right){x}^{2}+\left(-2a-4\right){x}-2a^{2}+a+9$ |
| 21.1-b1 |
21.1-b |
$1$ |
$1$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3 \cdot 7^{3} \) |
$4.09450$ |
$(-a-1), (a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$56.32716596$ |
2.041858171 |
\( -\frac{299328626099}{1029} a^{2} + \frac{50138437570}{147} a + \frac{1735402838198}{1029} \) |
\( \bigl[1\) , \( -a - 1\) , \( 0\) , \( 4 a\) , \( -2 a^{2} + 3 a + 1\bigr] \) |
${y}^2+{x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+4a{x}-2a^{2}+3a+1$ |
| 21.1-c1 |
21.1-c |
$1$ |
$1$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3 \cdot 7^{3} \) |
$4.09450$ |
$(-a-1), (a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.338775590$ |
$58.27186094$ |
2.146841219 |
\( -\frac{299328626099}{1029} a^{2} + \frac{50138437570}{147} a + \frac{1735402838198}{1029} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + 2 a + 4\) , \( a\) , \( 5 a^{2} - 16\) , \( 21 a^{2} - 15 a - 102\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+a{y}={x}^{3}+\left(-a^{2}+2a+4\right){x}^{2}+\left(5a^{2}-16\right){x}+21a^{2}-15a-102$ |
| 21.1-d1 |
21.1-d |
$6$ |
$8$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{2} \cdot 7^{4} \) |
$4.09450$ |
$(-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$64$ |
\( 2^{2} \) |
$1$ |
$0.966952621$ |
2.243328329 |
\( -\frac{6753481671027081357094325833}{21609} a^{2} + \frac{1131188974087934247922313456}{3087} a + \frac{39155136968953218755956232668}{21609} \) |
\( \bigl[a^{2} - 4\) , \( 1\) , \( a^{2} - 4\) , \( 230 a^{2} - 280 a - 1400\) , \( 3696 a^{2} - 4620 a - 22110\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+{x}^{2}+\left(230a^{2}-280a-1400\right){x}+3696a^{2}-4620a-22110$ |
| 21.1-d2 |
21.1-d |
$6$ |
$8$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{2} \cdot 7^{16} \) |
$4.09450$ |
$(-a-1), (a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$64$ |
\( 2^{2} \) |
$1$ |
$0.966952621$ |
2.243328329 |
\( \frac{5050883273717386823929705}{299096375126409} a^{2} + \frac{1489574000087096330787088}{42728053589487} a + \frac{1647849108085064735158004}{299096375126409} \) |
\( \bigl[a^{2} - 4\) , \( 1\) , \( a^{2} - 4\) , \( -140 a^{2} - 350 a - 140\) , \( -3416 a^{2} - 7322 a - 1596\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+{x}^{2}+\left(-140a^{2}-350a-140\right){x}-3416a^{2}-7322a-1596$ |
| 21.1-d3 |
21.1-d |
$6$ |
$8$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{8} \) |
$4.09450$ |
$(-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$16$ |
\( 2^{3} \) |
$1$ |
$7.735620971$ |
2.243328329 |
\( -\frac{110650233343082241553}{466948881} a^{2} + \frac{18533606397598137617}{66706983} a + \frac{641524754970175127041}{466948881} \) |
\( \bigl[a^{2} - 4\) , \( 1\) , \( a^{2} - 4\) , \( 5 a^{2} - 35 a - 90\) , \( 4 a^{2} - 245 a - 477\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+{x}^{2}+\left(5a^{2}-35a-90\right){x}+4a^{2}-245a-477$ |
| 21.1-d4 |
21.1-d |
$6$ |
$8$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{8} \cdot 7^{4} \) |
$4.09450$ |
$(-a-1), (a-1)$ |
0 |
$\Z/2\Z\oplus\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$4$ |
\( 2^{4} \) |
$1$ |
$61.88496777$ |
2.243328329 |
\( \frac{582023899160}{15752961} a^{2} - \frac{1038165632761}{2250423} a + \frac{16832160366340}{15752961} \) |
\( \bigl[a^{2} - 4\) , \( 1\) , \( a^{2} - 4\) , \( -5\) , \( a^{2} - 7 a - 14\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+{x}^{2}-5{x}+a^{2}-7a-14$ |
| 21.1-d5 |
21.1-d |
$6$ |
$8$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( 3^{4} \cdot 7^{2} \) |
$4.09450$ |
$(-a-1), (a-1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{3} \) |
$1$ |
$123.7699355$ |
2.243328329 |
\( \frac{40304401}{3969} a^{2} + \frac{11995048}{567} a + \frac{11325248}{3969} \) |
\( \bigl[a^{2} - 4\) , \( 1\) , \( a^{2} - 4\) , \( 0\) , \( 0\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+{x}^{2}$ |
| 21.1-d6 |
21.1-d |
$6$ |
$8$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
21.1 |
\( 3 \cdot 7 \) |
\( - 3^{16} \cdot 7^{2} \) |
$4.09450$ |
$(-a-1), (a-1)$ |
0 |
$\Z/8\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{5} \) |
$1$ |
$30.94248388$ |
2.243328329 |
\( \frac{1063395459790548094577}{2109289329} a^{2} - \frac{439327317478130560369}{301327047} a - \frac{562061870493043570577}{2109289329} \) |
\( \bigl[a^{2} - 4\) , \( 1\) , \( a^{2} - 4\) , \( -5 a^{2} + 35 a\) , \( 42 a^{2} - 77 a - 27\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+{x}^{2}+\left(-5a^{2}+35a\right){x}+42a^{2}-77a-27$ |
| 23.2-a1 |
23.2-a |
$1$ |
$1$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
23.2 |
\( 23 \) |
\( -23 \) |
$4.15705$ |
$(a^2-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$41.27345117$ |
1.496161436 |
\( -\frac{1812432}{23} a^{2} + \frac{2863257}{23} a + \frac{9076295}{23} \) |
\( \bigl[a^{2} - a - 3\) , \( -a - 1\) , \( a^{2} - 4\) , \( -a - 6\) , \( -a^{2} + 2 a - 4\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-a-6\right){x}-a^{2}+2a-4$ |
| 23.2-b1 |
23.2-b |
$1$ |
$1$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
23.2 |
\( 23 \) |
\( -23 \) |
$4.15705$ |
$(a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$0.066849611$ |
$314.3535004$ |
2.285315242 |
\( -\frac{1812432}{23} a^{2} + \frac{2863257}{23} a + \frac{9076295}{23} \) |
\( \bigl[1\) , \( a^{2} - a - 4\) , \( a^{2} - a - 4\) , \( 3 a^{2} - 3 a - 18\) , \( -12 a^{2} + 14 a + 69\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-a-4\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(3a^{2}-3a-18\right){x}-12a^{2}+14a+69$ |
| 23.2-c1 |
23.2-c |
$2$ |
$5$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
23.2 |
\( 23 \) |
\( - 23^{5} \) |
$4.15705$ |
$(a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.1 |
$1$ |
\( 1 \) |
$0.346350743$ |
$79.51304173$ |
2.994907528 |
\( -\frac{961139734845}{6436343} a^{2} + \frac{1173762893721}{6436343} a + \frac{5459934581087}{6436343} \) |
\( \bigl[a^{2} - a - 4\) , \( -a - 1\) , \( 0\) , \( -2 a^{2} + 7 a - 1\) , \( 5 a^{2} - 12 a - 10\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-2a^{2}+7a-1\right){x}+5a^{2}-12a-10$ |
| 23.2-c2 |
23.2-c |
$2$ |
$5$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
23.2 |
\( 23 \) |
\( -23 \) |
$4.15705$ |
$(a^2-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.4.2 |
$1$ |
\( 1 \) |
$1.731753715$ |
$15.90260834$ |
2.994907528 |
\( -\frac{90444121625531417484414}{23} a^{2} + \frac{261560279167054112306301}{23} a + \frac{47804598082098941998736}{23} \) |
\( \bigl[a^{2} - a - 4\) , \( -a - 1\) , \( 0\) , \( -1617 a^{2} + 4672 a + 719\) , \( 73180 a^{2} - 211965 a - 38243\bigr] \) |
${y}^2+\left(a^{2}-a-4\right){x}{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1617a^{2}+4672a+719\right){x}+73180a^{2}-211965a-38243$ |
| 23.2-d1 |
23.2-d |
$2$ |
$5$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
23.2 |
\( 23 \) |
\( -23 \) |
$4.15705$ |
$(a^2-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.2 |
$25$ |
\( 1 \) |
$1$ |
$0.417196637$ |
0.378084157 |
\( -\frac{90444121625531417484414}{23} a^{2} + \frac{261560279167054112306301}{23} a + \frac{47804598082098941998736}{23} \) |
\( \bigl[a^{2} - 4\) , \( a\) , \( a + 1\) , \( 374 a^{2} + 347 a - 4567\) , \( 12103 a^{2} + 4128 a - 126350\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(374a^{2}+347a-4567\right){x}+12103a^{2}+4128a-126350$ |
| 23.2-d2 |
23.2-d |
$2$ |
$5$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
23.2 |
\( 23 \) |
\( - 23^{5} \) |
$4.15705$ |
$(a^2-2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 5 \) |
$1$ |
$52.14957970$ |
0.378084157 |
\( -\frac{961139734845}{6436343} a^{2} + \frac{1173762893721}{6436343} a + \frac{5459934581087}{6436343} \) |
\( \bigl[a^{2} - 4\) , \( a\) , \( a + 1\) , \( 29 a^{2} - 28 a - 162\) , \( -105 a^{2} + 131 a + 621\bigr] \) |
${y}^2+\left(a^{2}-4\right){x}{y}+\left(a+1\right){y}={x}^{3}+a{x}^{2}+\left(29a^{2}-28a-162\right){x}-105a^{2}+131a+621$ |
| 27.1-a1 |
27.1-a |
$4$ |
$6$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{8} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.493631136$ |
$38.25328180$ |
3.080288759 |
\( -\frac{7670941654}{729} a^{2} + \frac{9002535971}{729} a + \frac{44448279379}{729} \) |
\( \bigl[1\) , \( a^{2} - 4\) , \( a\) , \( -a^{2} + 5 a + 3\) , \( -2 a^{2} + 6 a + 4\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-a^{2}+5a+3\right){x}-2a^{2}+6a+4$ |
| 27.1-a2 |
27.1-a |
$4$ |
$6$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{16} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.246815568$ |
$19.12664090$ |
3.080288759 |
\( \frac{214141345094741}{531441} a^{2} - \frac{619272725975563}{531441} a - \frac{113178315267704}{531441} \) |
\( \bigl[1\) , \( a^{2} - 4\) , \( a\) , \( -21 a^{2} + 60 a + 13\) , \( -107 a^{2} + 301 a + 58\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-21a^{2}+60a+13\right){x}-107a^{2}+301a+58$ |
| 27.1-a3 |
27.1-a |
$4$ |
$6$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{16} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{3} \cdot 3 \) |
$0.082271856$ |
$57.37992270$ |
3.080288759 |
\( -\frac{72423841}{729} a^{2} + \frac{89207618}{729} a + \frac{143253899}{243} \) |
\( \bigl[1\) , \( a + 1\) , \( a^{2} - 3\) , \( -844255 a^{2} - 1742908 a - 275499\) , \( 1161532765 a^{2} + 2397908351 a + 379036571\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(-844255a^{2}-1742908a-275499\right){x}+1161532765a^{2}+2397908351a+379036571$ |
| 27.1-a4 |
27.1-a |
$4$ |
$6$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{8} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \cdot 3 \) |
$0.164543712$ |
$114.7598454$ |
3.080288759 |
\( -\frac{4825}{27} a^{2} - \frac{1007}{27} a + \frac{48506}{27} \) |
\( \bigl[1\) , \( a + 1\) , \( a^{2} - 3\) , \( 15610 a^{2} + 32227 a + 5096\) , \( 61958378 a^{2} + 127909014 a + 20218533\bigr] \) |
${y}^2+{x}{y}+\left(a^{2}-3\right){y}={x}^{3}+\left(a+1\right){x}^{2}+\left(15610a^{2}+32227a+5096\right){x}+61958378a^{2}+127909014a+20218533$ |
| 27.1-b1 |
27.1-b |
$4$ |
$4$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{8} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$132.3584640$ |
1.799246465 |
\( -\frac{7318460659}{27} a^{2} + \frac{8584703501}{27} a + 1571060777 \) |
\( \bigl[a\) , \( a^{2} - a - 4\) , \( a^{2} - a - 3\) , \( -43 a^{2} + 124 a + 27\) , \( 283 a^{2} - 817 a - 153\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-43a^{2}+124a+27\right){x}+283a^{2}-817a-153$ |
| 27.1-b2 |
27.1-b |
$4$ |
$4$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{32} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \cdot 3 \) |
$1$ |
$8.272404003$ |
1.799246465 |
\( \frac{4879442218780}{531441} a^{2} + \frac{9849328391773}{531441} a + \frac{518348559730}{177147} \) |
\( \bigl[a\) , \( a^{2} - a - 4\) , \( a^{2} - a - 3\) , \( -703 a^{2} + 1954 a + 362\) , \( 19549 a^{2} - 57137 a - 10440\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-703a^{2}+1954a+362\right){x}+19549a^{2}-57137a-10440$ |
| 27.1-b3 |
27.1-b |
$4$ |
$4$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{16} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \cdot 3 \) |
$1$ |
$66.17923203$ |
1.799246465 |
\( \frac{45147939170713}{729} a^{2} - \frac{130565789194484}{729} a - \frac{7954364645372}{243} \) |
\( \bigl[a\) , \( a^{2} - a - 4\) , \( a^{2} - a - 3\) , \( -688 a^{2} + 1984 a + 367\) , \( 19682 a^{2} - 56933 a - 10409\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-688a^{2}+1984a+367\right){x}+19682a^{2}-56933a-10409$ |
| 27.1-b4 |
27.1-b |
$4$ |
$4$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{8} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$33.08961601$ |
1.799246465 |
\( \frac{882563049076772542022860}{27} a^{2} - \frac{2552332128943458364490687}{27} a - 17277118113217461091814 \) |
\( \bigl[a\) , \( a^{2} - a - 4\) , \( a^{2} - a - 3\) , \( -10993 a^{2} + 31774 a + 5812\) , \( 1285811 a^{2} - 3718493 a - 679622\bigr] \) |
${y}^2+a{x}{y}+\left(a^{2}-a-3\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-10993a^{2}+31774a+5812\right){x}+1285811a^{2}-3718493a-679622$ |
| 27.1-c1 |
27.1-c |
$1$ |
$1$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{7} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 1 \) |
$1$ |
$63.69653089$ |
2.308997440 |
\( -\frac{544768}{27} a^{2} - \frac{1134592}{27} a - \frac{65536}{9} \) |
\( \bigl[0\) , \( a^{2} - a - 4\) , \( a^{2} - 3\) , \( -a^{2} + 2 a + 5\) , \( 26 a^{2} - 77 a - 18\bigr] \) |
${y}^2+\left(a^{2}-3\right){y}={x}^{3}+\left(a^{2}-a-4\right){x}^{2}+\left(-a^{2}+2a+5\right){x}+26a^{2}-77a-18$ |
| 27.1-d1 |
27.1-d |
$1$ |
$1$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{7} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 3 \) |
$1$ |
$15.95730316$ |
1.735355362 |
\( -\frac{544768}{27} a^{2} - \frac{1134592}{27} a - \frac{65536}{9} \) |
\( \bigl[0\) , \( -a^{2} + 2 a + 5\) , \( a + 1\) , \( 1\) , \( -a^{2} + 7 a - 16\bigr] \) |
${y}^2+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2a+5\right){x}^{2}+{x}-a^{2}+7a-16$ |
| 27.1-e1 |
27.1-e |
$4$ |
$4$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{8} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$16$ |
\( 2 \) |
$1$ |
$9.159920878$ |
2.656374979 |
\( -\frac{7318460659}{27} a^{2} + \frac{8584703501}{27} a + 1571060777 \) |
\( \bigl[a^{2} - a - 3\) , \( -a^{2} + 2 a + 3\) , \( a + 1\) , \( -12 a^{2} + 33 a + 7\) , \( -33 a^{2} + 99 a + 3\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2a+3\right){x}^{2}+\left(-12a^{2}+33a+7\right){x}-33a^{2}+99a+3$ |
| 27.1-e2 |
27.1-e |
$4$ |
$4$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{32} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$4$ |
\( 2^{4} \) |
$1$ |
$4.579960439$ |
2.656374979 |
\( \frac{4879442218780}{531441} a^{2} + \frac{9849328391773}{531441} a + \frac{518348559730}{177147} \) |
\( \bigl[a^{2} - a - 3\) , \( -a^{2} + 2 a + 3\) , \( a + 1\) , \( -192 a^{2} + 548 a + 92\) , \( -2807 a^{2} + 8095 a + 1473\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2a+3\right){x}^{2}+\left(-192a^{2}+548a+92\right){x}-2807a^{2}+8095a+1473$ |
| 27.1-e3 |
27.1-e |
$4$ |
$4$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{16} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
$0 \le r \le 2$ |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$16$ |
\( 2^{3} \) |
$1$ |
$9.159920878$ |
2.656374979 |
\( \frac{45147939170713}{729} a^{2} - \frac{130565789194484}{729} a - \frac{7954364645372}{243} \) |
\( \bigl[a^{2} - a - 3\) , \( -a^{2} + 2 a + 3\) , \( a + 1\) , \( -192 a^{2} + 553 a + 102\) , \( -2803 a^{2} + 8109 a + 1467\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2a+3\right){x}^{2}+\left(-192a^{2}+553a+102\right){x}-2803a^{2}+8109a+1467$ |
| 27.1-e4 |
27.1-e |
$4$ |
$4$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( - 3^{8} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
$0 \le r \le 2$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$64$ |
\( 2 \) |
$1$ |
$2.289980219$ |
2.656374979 |
\( \frac{882563049076772542022860}{27} a^{2} - \frac{2552332128943458364490687}{27} a - 17277118113217461091814 \) |
\( \bigl[a^{2} - a - 3\) , \( -a^{2} + 2 a + 3\) , \( a + 1\) , \( -3072 a^{2} + 8878 a + 1632\) , \( -188239 a^{2} + 544383 a + 99477\bigr] \) |
${y}^2+\left(a^{2}-a-3\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(-a^{2}+2a+3\right){x}^{2}+\left(-3072a^{2}+8878a+1632\right){x}-188239a^{2}+544383a+99477$ |
| 27.1-f1 |
27.1-f |
$4$ |
$6$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{8} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.766276409$ |
$46.96262337$ |
1.956756275 |
\( -\frac{7670941654}{729} a^{2} + \frac{9002535971}{729} a + \frac{44448279379}{729} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + 5\) , \( a^{2} - 4\) , \( 13 a^{2} - 16 a - 79\) , \( 92 a^{2} - 108 a - 533\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+5\right){x}^{2}+\left(13a^{2}-16a-79\right){x}+92a^{2}-108a-533$ |
| 27.1-f2 |
27.1-f |
$4$ |
$6$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{16} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.383138204$ |
$46.96262337$ |
1.956756275 |
\( \frac{214141345094741}{531441} a^{2} - \frac{619272725975563}{531441} a - \frac{113178315267704}{531441} \) |
\( \bigl[a^{2} - 3\) , \( -a^{2} + 5\) , \( a^{2} - 4\) , \( 8 a^{2} - a - 79\) , \( 102 a^{2} - 142 a - 525\bigr] \) |
${y}^2+\left(a^{2}-3\right){x}{y}+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+5\right){x}^{2}+\left(8a^{2}-a-79\right){x}+102a^{2}-142a-525$ |
| 27.1-f3 |
27.1-f |
$4$ |
$6$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{16} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2^{2} \) |
$0.127712734$ |
$140.8878701$ |
1.956756275 |
\( -\frac{72423841}{729} a^{2} + \frac{89207618}{729} a + \frac{143253899}{243} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -6 a - 11\) , \( a^{2} + 9 a + 13\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-6a-11\right){x}+a^{2}+9a+13$ |
| 27.1-f4 |
27.1-f |
$4$ |
$6$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.1 |
\( 3^{3} \) |
\( 3^{8} \) |
$4.26964$ |
$(-a-1), (a^2-2a-4)$ |
$1$ |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2, 3$ |
2B, 3B |
$1$ |
\( 2 \) |
$0.255425469$ |
$140.8878701$ |
1.956756275 |
\( -\frac{4825}{27} a^{2} - \frac{1007}{27} a + \frac{48506}{27} \) |
\( \bigl[a\) , \( -a\) , \( a\) , \( -a - 1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^{3}-a{x}^{2}+\left(-a-1\right){x}$ |
| 27.2-a1 |
27.2-a |
$2$ |
$3$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{9} \) |
$4.26964$ |
$(-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 3 \) |
$1$ |
$19.78405596$ |
2.151514405 |
\( 102846464 a^{2} - 297046016 a - 54292480 \) |
\( \bigl[0\) , \( a^{2} - 2 a - 3\) , \( a^{2} - 4\) , \( -12 a^{2} + 24 a + 39\) , \( -20 a^{2} + 70 a - 34\bigr] \) |
${y}^2+\left(a^{2}-4\right){y}={x}^{3}+\left(a^{2}-2a-3\right){x}^{2}+\left(-12a^{2}+24a+39\right){x}-20a^{2}+70a-34$ |
| 27.2-a2 |
27.2-a |
$2$ |
$3$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{11} \) |
$4.26964$ |
$(-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$1$ |
$59.35216788$ |
2.151514405 |
\( 4096 a^{2} - 4096 a - 20480 \) |
\( \bigl[0\) , \( -a^{2} + 2 a + 3\) , \( a^{2} - 4\) , \( -6916 a^{2} - 14280 a - 2255\) , \( 732885 a^{2} + 1512993 a + 239155\bigr] \) |
${y}^2+\left(a^{2}-4\right){y}={x}^{3}+\left(-a^{2}+2a+3\right){x}^{2}+\left(-6916a^{2}-14280a-2255\right){x}+732885a^{2}+1512993a+239155$ |
| 27.2-b1 |
27.2-b |
$2$ |
$3$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{11} \) |
$4.26964$ |
$(-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$0.135441050$ |
$204.7775816$ |
3.016210508 |
\( 4096 a^{2} - 4096 a - 20480 \) |
\( \bigl[0\) , \( a^{2} - 4\) , \( 1\) , \( -64950 a^{2} - 134082 a - 21189\) , \( 20956423 a^{2} + 43263162 a + 6838591\bigr] \) |
${y}^2+{y}={x}^{3}+\left(a^{2}-4\right){x}^{2}+\left(-64950a^{2}-134082a-21189\right){x}+20956423a^{2}+43263162a+6838591$ |
| 27.2-b2 |
27.2-b |
$2$ |
$3$ |
3.3.761.1 |
$3$ |
$[3, 0]$ |
27.2 |
\( 3^{3} \) |
\( - 3^{9} \) |
$4.26964$ |
$(-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
✓ |
$3$ |
3B |
$1$ |
\( 1 \) |
$0.406323150$ |
$68.25919387$ |
3.016210508 |
\( 102846464 a^{2} - 297046016 a - 54292480 \) |
\( \bigl[0\) , \( -a^{2} + 4\) , \( 1\) , \( -24 a^{2} + 70 a + 19\) , \( 123 a^{2} - 358 a - 62\bigr] \) |
${y}^2+{y}={x}^{3}+\left(-a^{2}+4\right){x}^{2}+\left(-24a^{2}+70a+19\right){x}+123a^{2}-358a-62$ |
*The rank, regulator and analytic order of Ш are
not known for all curves in the database; curves for which these are
unknown will not appear in searches specifying one of these
quantities.
