| 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 |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$0.81321$ |
$(-2a+1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cn, 3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$0.935309008$ |
0.858298410 |
\( -\frac{50357871050752}{19} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -769\) , \( -8470\bigr] \) |
${y}^2+{y}={x}^3+{x}^2-769{x}-8470$ |
| 19.1-a2 |
19.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 19^{6} \) |
$0.81321$ |
$(-2a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cn, 3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$2.805927025$ |
0.858298410 |
\( -\frac{89915392}{6859} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -9\) , \( -15\bigr] \) |
${y}^2+{y}={x}^3+{x}^2-9{x}-15$ |
| 19.1-a3 |
19.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
19.1 |
\( 19 \) |
\( 19^{2} \) |
$0.81321$ |
$(-2a+1)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 3$ |
2Cn, 3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$8.417781075$ |
0.858298410 |
\( \frac{32768}{19} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 1\) , \( 0\bigr] \) |
${y}^2+{y}={x}^3+{x}^2+{x}$ |
| 20.1-a1 |
20.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{2} \cdot 5^{18} \) |
$0.82371$ |
$(-a), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$1.041796112$ |
0.956017679 |
\( \frac{546495468563548}{3814697265625} a - \frac{26594457793024591}{7629394531250} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -15 a - 31\) , \( -54 a - 60\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2+\left(-15a-31\right){x}-54a-60$ |
| 20.1-a2 |
20.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{2} \cdot 5^{2} \) |
$0.82371$ |
$(-a), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$9.376165015$ |
0.956017679 |
\( -\frac{16129}{50} a + \frac{10942}{25} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -1\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2-{x}$ |
| 20.1-a3 |
20.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
20.1 |
\( 2^{2} \cdot 5 \) |
\( 2^{6} \cdot 5^{6} \) |
$0.82371$ |
$(-a), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$3.125388338$ |
0.956017679 |
\( -\frac{1697253523}{15625} a + \frac{9034902289}{125000} \) |
\( \bigl[a\) , \( -a + 1\) , \( 1\) , \( -5 a + 9\) , \( -2 a - 24\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(-a+1\right){x}^2+\left(-5a+9\right){x}-2a-24$ |
| 20.2-a1 |
20.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( 2^{6} \cdot 5^{6} \) |
$0.82371$ |
$(a-1), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$1$ |
$3.125388338$ |
0.956017679 |
\( \frac{1697253523}{15625} a - \frac{908625179}{25000} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 4 a + 4\) , \( 2 a - 26\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(4a+4\right){x}+2a-26$ |
| 20.2-a2 |
20.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( 2^{2} \cdot 5^{18} \) |
$0.82371$ |
$(a-1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$1.041796112$ |
0.956017679 |
\( -\frac{546495468563548}{3814697265625} a - \frac{5100293371179499}{1525878906250} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( 14 a - 46\) , \( 54 a - 114\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(14a-46\right){x}+54a-114$ |
| 20.2-a3 |
20.2-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
20.2 |
\( 2^{2} \cdot 5 \) |
\( 2^{2} \cdot 5^{2} \) |
$0.82371$ |
$(a-1), (2)$ |
0 |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \) |
$1$ |
$9.376165015$ |
0.956017679 |
\( \frac{16129}{50} a + \frac{1151}{10} \) |
\( \bigl[a + 1\) , \( 0\) , \( 1\) , \( -a - 1\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+{y}={x}^3+\left(-a-1\right){x}$ |
| 44.1-a1 |
44.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
44.1 |
\( 2^{2} \cdot 11 \) |
\( 2^{14} \cdot 11^{2} \) |
$1.00318$ |
$(a+2), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.157781648$ |
$4.016441512$ |
1.163084113 |
\( -\frac{1436117}{15488} a + \frac{10855315}{3872} \) |
\( \bigl[1\) , \( 0\) , \( a + 1\) , \( a - 1\) , \( -a\bigr] \) |
${y}^2+{x}{y}+\left(a+1\right){y}={x}^3+\left(a-1\right){x}-a$ |
| 44.2-a1 |
44.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
44.2 |
\( 2^{2} \cdot 11 \) |
\( 2^{14} \cdot 11^{2} \) |
$1.00318$ |
$(a-3), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
|
|
$1$ |
\( 2 \) |
$0.157781648$ |
$4.016441512$ |
1.163084113 |
\( \frac{1436117}{15488} a + \frac{41985143}{15488} \) |
\( \bigl[1\) , \( 0\) , \( a\) , \( -2 a + 1\) , \( 0\bigr] \) |
${y}^2+{x}{y}+a{y}={x}^3+\left(-2a+1\right){x}$ |
| 49.3-CMa1 |
49.3-CMa |
$1$ |
$1$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
49.3 |
\( 7^{2} \) |
\( 7^{6} \) |
$1.03054$ |
$(a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{yes}$ |
$-19$ |
$\mathrm{U}(1)$ |
✓ |
|
|
✓ |
$7$ |
7Cs.2.1 |
$1$ |
\( 1 \) |
$1$ |
$3.318662149$ |
1.522706625 |
\( -884736 \) |
\( \bigl[0\) , \( 0\) , \( 1\) , \( -6 a - 2\) , \( -9 a + 9\bigr] \) |
${y}^2+{y}={x}^3+\left(-6a-2\right){x}-9a+9$ |
| 76.1-a1 |
76.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
76.1 |
\( 2^{2} \cdot 19 \) |
\( 2^{2} \cdot 19^{10} \) |
$1.15006$ |
$(-2a+1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 2 \) |
$1$ |
$0.965055962$ |
0.885596087 |
\( -\frac{37966934881}{4952198} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( -70\) , \( -279\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2-70{x}-279$ |
| 76.1-a2 |
76.1-a |
$2$ |
$5$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
76.1 |
\( 2^{2} \cdot 19 \) |
\( 2^{10} \cdot 19^{2} \) |
$1.15006$ |
$(-2a+1), (2)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 2 \cdot 5 \) |
$1$ |
$4.825279813$ |
0.885596087 |
\( -\frac{1}{608} \) |
\( \bigl[1\) , \( 1\) , \( 1\) , \( 0\) , \( 1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+{x}^2+1$ |
| 76.1-b1 |
76.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
76.1 |
\( 2^{2} \cdot 19 \) |
\( 2^{6} \cdot 19^{2} \) |
$1.15006$ |
$(-2a+1), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.1 |
$1$ |
\( 2 \cdot 3 \) |
$0.774675563$ |
$3.410590199$ |
1.616372036 |
\( -\frac{413493625}{152} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -16\) , \( 22\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-16{x}+22$ |
| 76.1-b2 |
76.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
76.1 |
\( 2^{2} \cdot 19 \) |
\( 2^{54} \cdot 19^{2} \) |
$1.15006$ |
$(-2a+1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3B.1.2 |
$1$ |
\( 2 \cdot 3^{3} \) |
$0.086075062$ |
$0.378954466$ |
1.616372036 |
\( -\frac{69173457625}{2550136832} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -86\) , \( -2456\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-86{x}-2456$ |
| 76.1-b3 |
76.1-b |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
76.1 |
\( 2^{2} \cdot 19 \) |
\( 2^{18} \cdot 19^{6} \) |
$1.15006$ |
$(-2a+1), (2)$ |
$1$ |
$\Z/3\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$3$ |
3Cs.1.1 |
$1$ |
\( 2 \cdot 3^{3} \) |
$0.258225187$ |
$1.136863399$ |
1.616372036 |
\( \frac{94196375}{3511808} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 9\) , \( 90\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+9{x}+90$ |
| 85.2-a1 |
85.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
85.2 |
\( 5 \cdot 17 \) |
\( 5^{4} \cdot 17^{2} \) |
$1.18269$ |
$(-a), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.115004555$ |
1.888093579 |
\( \frac{3619812216}{180625} a - \frac{3820287361}{180625} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( 4\) , \( -4 a + 1\bigr] \) |
${y}^2+{x}{y}={x}^3-a{x}^2+4{x}-4a+1$ |
| 85.2-a2 |
85.2-a |
$2$ |
$2$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
85.2 |
\( 5 \cdot 17 \) |
\( 5^{2} \cdot 17 \) |
$1.18269$ |
$(-a), (a-4)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.230009110$ |
1.888093579 |
\( -\frac{205232}{425} a + \frac{620697}{425} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^3-a{x}^2-{x}$ |
| 85.3-a1 |
85.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
85.3 |
\( 5 \cdot 17 \) |
\( 5^{4} \cdot 17^{2} \) |
$1.18269$ |
$(a-1), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{2} \) |
$1$ |
$4.115004555$ |
1.888093579 |
\( -\frac{3619812216}{180625} a - \frac{40095029}{36125} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 4\) , \( 4 a - 3\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a-1\right){x}^2+4{x}+4a-3$ |
| 85.3-a2 |
85.3-a |
$2$ |
$2$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
85.3 |
\( 5 \cdot 17 \) |
\( 5^{2} \cdot 17 \) |
$1.18269$ |
$(a-1), (a+3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$8.230009110$ |
1.888093579 |
\( \frac{205232}{425} a + \frac{83093}{85} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -1\) , \( 0\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a-1\right){x}^2-{x}$ |
| 100.1-a1 |
100.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{2} \cdot 5^{24} \) |
$1.23173$ |
$(-a), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1.941303345$ |
$0.465905385$ |
1.659985596 |
\( \frac{546495468563548}{3814697265625} a - \frac{26594457793024591}{7629394531250} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 29 a + 223\) , \( -697 a + 1006\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a-1\right){x}^2+\left(29a+223\right){x}-697a+1006$ |
| 100.1-a2 |
100.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{2} \cdot 5^{8} \) |
$1.23173$ |
$(-a), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.215700371$ |
$4.193148468$ |
1.659985596 |
\( -\frac{16129}{50} a + \frac{10942}{25} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( -a - 2\) , \( -a + 1\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a-1\right){x}^2+\left(-a-2\right){x}-a+1$ |
| 100.1-a3 |
100.1-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
100.1 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{6} \cdot 5^{12} \) |
$1.23173$ |
$(-a), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$0.647101115$ |
$1.397716156$ |
1.659985596 |
\( -\frac{1697253523}{15625} a + \frac{9034902289}{125000} \) |
\( \bigl[1\) , \( a - 1\) , \( 0\) , \( 29 a - 27\) , \( -85 a - 104\bigr] \) |
${y}^2+{x}{y}={x}^3+\left(a-1\right){x}^2+\left(29a-27\right){x}-85a-104$ |
| 100.3-a1 |
100.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
100.3 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{6} \cdot 5^{12} \) |
$1.23173$ |
$(a-1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cs |
$1$ |
\( 2 \) |
$0.647101115$ |
$1.397716156$ |
1.659985596 |
\( \frac{1697253523}{15625} a - \frac{908625179}{25000} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( -29 a + 2\) , \( 85 a - 189\bigr] \) |
${y}^2+{x}{y}={x}^3-a{x}^2+\left(-29a+2\right){x}+85a-189$ |
| 100.3-a2 |
100.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
100.3 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{2} \cdot 5^{24} \) |
$1.23173$ |
$(a-1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$1.941303345$ |
$0.465905385$ |
1.659985596 |
\( -\frac{546495468563548}{3814697265625} a - \frac{5100293371179499}{1525878906250} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( -29 a + 252\) , \( 697 a + 309\bigr] \) |
${y}^2+{x}{y}={x}^3-a{x}^2+\left(-29a+252\right){x}+697a+309$ |
| 100.3-a3 |
100.3-a |
$3$ |
$9$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
100.3 |
\( 2^{2} \cdot 5^{2} \) |
\( 2^{2} \cdot 5^{8} \) |
$1.23173$ |
$(a-1), (2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3B |
$1$ |
\( 2 \) |
$0.215700371$ |
$4.193148468$ |
1.659985596 |
\( \frac{16129}{50} a + \frac{1151}{10} \) |
\( \bigl[1\) , \( -a\) , \( 0\) , \( a - 3\) , \( a\bigr] \) |
${y}^2+{x}{y}={x}^3-a{x}^2+\left(a-3\right){x}+a$ |
| 121.2-a1 |
121.2-a |
$3$ |
$25$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
121.2 |
\( 11^{2} \) |
\( 11^{2} \) |
$1.29185$ |
$(a+2), (a-3)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.2 |
$1$ |
\( 1 \) |
$5.612837583$ |
$0.370308724$ |
1.907346561 |
\( -\frac{52893159101157376}{11} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -7820\) , \( -263580\bigr] \) |
${y}^2+{y}={x}^3-{x}^2-7820{x}-263580$ |
| 121.2-a2 |
121.2-a |
$3$ |
$25$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
121.2 |
\( 11^{2} \) |
\( 11^{10} \) |
$1.29185$ |
$(a+2), (a-3)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5Cs.1.1 |
$1$ |
\( 5^{2} \) |
$1.122567516$ |
$1.851543623$ |
1.907346561 |
\( -\frac{122023936}{161051} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -10\) , \( -20\bigr] \) |
${y}^2+{y}={x}^3-{x}^2-10{x}-20$ |
| 121.2-a3 |
121.2-a |
$3$ |
$25$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
121.2 |
\( 11^{2} \) |
\( 11^{2} \) |
$1.29185$ |
$(a+2), (a-3)$ |
$1$ |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$5$ |
5B.1.1 |
$1$ |
\( 1 \) |
$5.612837583$ |
$9.257718117$ |
1.907346561 |
\( -\frac{4096}{11} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+{y}={x}^3-{x}^2$ |
| 171.1-a1 |
171.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{2} \cdot 19^{8} \) |
$1.40852$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$1.632344499$ |
0.374485511 |
\( \frac{67419143}{390963} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( 8\) , \( 29\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3+8{x}+29$ |
| 171.1-a2 |
171.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{2} \cdot 19^{2} \) |
$1.40852$ |
$(-2a+1), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$6.529377996$ |
0.374485511 |
\( \frac{389017}{57} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -2\) , \( -1\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-2{x}-1$ |
| 171.1-a3 |
171.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{4} \cdot 19^{4} \) |
$1.40852$ |
$(-2a+1), (3)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{2} \) |
$1$ |
$3.264688998$ |
0.374485511 |
\( \frac{30664297}{3249} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -7\) , \( 5\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-7{x}+5$ |
| 171.1-a4 |
171.1-a |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{8} \cdot 19^{2} \) |
$1.40852$ |
$(-2a+1), (3)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.632344499$ |
0.374485511 |
\( \frac{115714886617}{1539} \) |
\( \bigl[1\) , \( 0\) , \( 1\) , \( -102\) , \( 385\bigr] \) |
${y}^2+{x}{y}+{y}={x}^3-102{x}+385$ |
| 171.1-b1 |
171.1-b |
$1$ |
$1$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{4} \cdot 19^{2} \) |
$1.40852$ |
$(-2a+1), (3)$ |
$2$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2$ |
2Cn |
$1$ |
\( 2^{2} \) |
$0.044869955$ |
$5.328644115$ |
1.755276488 |
\( -\frac{1404928}{171} \) |
\( \bigl[0\) , \( -1\) , \( 1\) , \( -2\) , \( 2\bigr] \) |
${y}^2+{y}={x}^3-{x}^2-2{x}+2$ |
| 171.1-c1 |
171.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{4} \cdot 19^{10} \) |
$1.40852$ |
$(-2a+1), (3)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2Cn, 5B.1.2 |
$4$ |
\( 2^{2} \) |
$1$ |
$0.271765830$ |
1.995115434 |
\( -\frac{9358714467168256}{22284891} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( -4390\) , \( -113432\bigr] \) |
${y}^2+{y}={x}^3+{x}^2-4390{x}-113432$ |
| 171.1-c2 |
171.1-c |
$2$ |
$5$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
171.1 |
\( 3^{2} \cdot 19 \) |
\( 3^{20} \cdot 19^{2} \) |
$1.40852$ |
$(-2a+1), (3)$ |
0 |
$\Z/5\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
✓ |
✓ |
|
$2, 5$ |
2Cn, 5B.1.1 |
$4$ |
\( 2^{2} \cdot 5 \) |
$1$ |
$1.358829150$ |
1.995115434 |
\( \frac{841232384}{1121931} \) |
\( \bigl[0\) , \( 1\) , \( 1\) , \( 20\) , \( -32\bigr] \) |
${y}^2+{y}={x}^3+{x}^2+20{x}-32$ |
| 175.2-a1 |
175.2-a |
$1$ |
$1$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
175.2 |
\( 5^{2} \cdot 7 \) |
\( 5^{3} \cdot 7^{3} \) |
$1.41669$ |
$(-a), (a-2)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cn |
$1$ |
\( 2 \cdot 3 \) |
$0.062767475$ |
$5.334923105$ |
1.843729767 |
\( -\frac{654642}{343} a + \frac{1310769}{343} \) |
\( \bigl[a + 1\) , \( 1\) , \( a\) , \( 0\) , \( 0\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+a{y}={x}^3+{x}^2$ |
| 175.2-b1 |
175.2-b |
$1$ |
$1$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
175.2 |
\( 5^{2} \cdot 7 \) |
\( 5^{9} \cdot 7^{3} \) |
$1.41669$ |
$(-a), (a-2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cn |
$1$ |
\( 2 \) |
$1$ |
$2.385850143$ |
2.189406246 |
\( -\frac{654642}{343} a + \frac{1310769}{343} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( -4 a + 7\) , \( 7 a - 18\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a-1\right){x}^2+\left(-4a+7\right){x}+7a-18$ |
| 175.3-a1 |
175.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
175.3 |
\( 5^{2} \cdot 7 \) |
\( 5^{4} \cdot 7^{4} \) |
$1.41669$ |
$(-a), (a-1), (-a-1)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$3.757058949$ |
1.723856872 |
\( \frac{21610347}{12005} a - \frac{741467764}{60025} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -a - 2\) , \( -a - 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a-2\right){x}-a-1$ |
| 175.3-a2 |
175.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
175.3 |
\( 5^{2} \cdot 7 \) |
\( 5^{2} \cdot 7^{2} \) |
$1.41669$ |
$(-a), (a-1), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$7.514117899$ |
1.723856872 |
\( \frac{72237}{245} a - \frac{228761}{245} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -a + 3\) , \( 1\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a+3\right){x}+1$ |
| 175.3-a3 |
175.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
175.3 |
\( 5^{2} \cdot 7 \) |
\( 5^{5} \cdot 7^{8} \) |
$1.41669$ |
$(-a), (a-1), (-a-1)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$1.878529474$ |
1.723856872 |
\( -\frac{2532825772081}{3603000625} a - \frac{1341942703599}{3603000625} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( 4 a - 7\) , \( -13 a + 6\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(4a-7\right){x}-13a+6$ |
| 175.3-a4 |
175.3-a |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
175.3 |
\( 5^{2} \cdot 7 \) |
\( 5^{5} \cdot 7^{2} \) |
$1.41669$ |
$(-a), (a-1), (-a-1)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.878529474$ |
1.723856872 |
\( -\frac{1344745702751}{30625} a + \frac{948946059331}{6125} \) |
\( \bigl[a\) , \( 0\) , \( a\) , \( -6 a - 77\) , \( -33 a - 236\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-6a-77\right){x}-33a-236$ |
| 175.4-a1 |
175.4-a |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
175.4 |
\( 5^{2} \cdot 7 \) |
\( 5^{4} \cdot 7^{4} \) |
$1.41669$ |
$(-a), (a-1), (a-2)$ |
0 |
$\Z/2\Z\oplus\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2Cs |
$1$ |
\( 2^{4} \) |
$1$ |
$3.757058949$ |
1.723856872 |
\( -\frac{21610347}{12005} a - \frac{633416029}{60025} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -a - 3\) , \( -2\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-a-3\right){x}-2$ |
| 175.4-a2 |
175.4-a |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
175.4 |
\( 5^{2} \cdot 7 \) |
\( 5^{5} \cdot 7^{8} \) |
$1.41669$ |
$(-a), (a-1), (a-2)$ |
0 |
$\Z/4\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{5} \) |
$1$ |
$1.878529474$ |
1.723856872 |
\( \frac{2532825772081}{3603000625} a - \frac{774953695136}{720600125} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -6 a - 3\) , \( 12 a - 7\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-6a-3\right){x}+12a-7$ |
| 175.4-a3 |
175.4-a |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
175.4 |
\( 5^{2} \cdot 7 \) |
\( 5^{2} \cdot 7^{2} \) |
$1.41669$ |
$(-a), (a-1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2 \) |
$1$ |
$7.514117899$ |
1.723856872 |
\( -\frac{72237}{245} a - \frac{156524}{245} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( -a + 2\) , \( -a + 1\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(-a+2\right){x}-a+1$ |
| 175.4-a4 |
175.4-a |
$4$ |
$4$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
175.4 |
\( 5^{2} \cdot 7 \) |
\( 5^{5} \cdot 7^{2} \) |
$1.41669$ |
$(-a), (a-1), (a-2)$ |
0 |
$\Z/2\Z$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
✓ |
|
$2$ |
2B |
$1$ |
\( 2^{3} \) |
$1$ |
$1.878529474$ |
1.723856872 |
\( \frac{1344745702751}{30625} a + \frac{3399984593904}{30625} \) |
\( \bigl[a + 1\) , \( -a\) , \( a + 1\) , \( 4 a - 83\) , \( 32 a - 269\bigr] \) |
${y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^3-a{x}^2+\left(4a-83\right){x}+32a-269$ |
| 175.5-a1 |
175.5-a |
$1$ |
$1$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
175.5 |
\( 5^{2} \cdot 7 \) |
\( 5^{3} \cdot 7^{3} \) |
$1.41669$ |
$(a-1), (-a-1)$ |
$1$ |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cn |
$1$ |
\( 2 \cdot 3 \) |
$0.062767475$ |
$5.334923105$ |
1.843729767 |
\( \frac{654642}{343} a + \frac{656127}{343} \) |
\( \bigl[a\) , \( -a - 1\) , \( 1\) , \( 0\) , \( 0\bigr] \) |
${y}^2+a{x}{y}+{y}={x}^3+\left(-a-1\right){x}^2$ |
| 175.5-b1 |
175.5-b |
$1$ |
$1$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
175.5 |
\( 5^{2} \cdot 7 \) |
\( 5^{9} \cdot 7^{3} \) |
$1.41669$ |
$(a-1), (-a-1)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
|
|
|
|
$3$ |
3Cn |
$1$ |
\( 2 \) |
$1$ |
$2.385850143$ |
2.189406246 |
\( \frac{654642}{343} a + \frac{656127}{343} \) |
\( \bigl[a + 1\) , \( 1\) , \( 0\) , \( 5 a - 2\) , \( -3 a - 13\bigr] \) |
${y}^2+\left(a+1\right){x}{y}={x}^3+{x}^2+\left(5a-2\right){x}-3a-13$ |
| 196.1-a1 |
196.1-a |
$1$ |
$1$ |
\(\Q(\sqrt{-19}) \) |
$2$ |
$[0, 1]$ |
196.1 |
\( 2^{2} \cdot 7^{2} \) |
\( 2^{6} \cdot 7^{6} \) |
$1.45740$ |
$(-a-1), (2)$ |
0 |
$\mathsf{trivial}$ |
$\textsf{no}$ |
|
$\mathrm{SU}(2)$ |
✓ |
|
|
|
$3$ |
3Cn |
$1$ |
\( 1 \) |
$1$ |
$3.751213062$ |
1.721174595 |
\( -\frac{27}{8} \) |
\( \bigl[a\) , \( -a - 1\) , \( a\) , \( a + 3\) , \( -2 a + 2\bigr] \) |
${y}^2+a{x}{y}+a{y}={x}^3+\left(-a-1\right){x}^2+\left(a+3\right){x}-2a+2$ |
*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.
