Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x + 1)y = 3x^3 + 4x^2 + x$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2 + z^3)y = 3x^3z^3 + 4x^2z^4 + xz^5$ | (dehomogenize, simplify) |
| $y^2 = x^6 + 2x^4 + 14x^3 + 17x^2 + 6x + 1$ | (homogenize, minimize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(21316\) | \(=\) | \( 2^{2} \cdot 73^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-42632\) | \(=\) | \( - 2^{3} \cdot 73^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(196\) | \(=\) | \( 2^{2} \cdot 7^{2} \) |
| \( I_4 \) | \(=\) | \(12337\) | \(=\) | \( 13^{2} \cdot 73 \) |
| \( I_6 \) | \(=\) | \(588745\) | \(=\) | \( 5 \cdot 73 \cdot 1613 \) |
| \( I_{10} \) | \(=\) | \(-5456896\) | \(=\) | \( - 2^{10} \cdot 73^{2} \) |
| \( J_2 \) | \(=\) | \(49\) | \(=\) | \( 7^{2} \) |
| \( J_4 \) | \(=\) | \(-414\) | \(=\) | \( - 2 \cdot 3^{2} \cdot 23 \) |
| \( J_6 \) | \(=\) | \(-908\) | \(=\) | \( - 2^{2} \cdot 227 \) |
| \( J_8 \) | \(=\) | \(-53972\) | \(=\) | \( - 2^{2} \cdot 103 \cdot 131 \) |
| \( J_{10} \) | \(=\) | \(-42632\) | \(=\) | \( - 2^{3} \cdot 73^{2} \) |
| \( g_1 \) | \(=\) | \(-282475249/42632\) | ||
| \( g_2 \) | \(=\) | \(24353343/21316\) | ||
| \( g_3 \) | \(=\) | \(545027/10658\) |
Automorphism group
| \(\mathrm{Aut}(X)\) | \(\simeq\) | $C_6$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_6$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
| Known points | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : 1 : 1)\) |
| \((-1 : 0 : 3)\) | \((2 : 3 : 1)\) | \((-3 : 7 : 2)\) | \((2 : -14 : 1)\) | \((-1 : -17 : 3)\) | \((-3 : 24 : 2)\) |
| Known points | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : 1 : 1)\) |
| \((-1 : 0 : 3)\) | \((2 : 3 : 1)\) | \((-3 : 7 : 2)\) | \((2 : -14 : 1)\) | \((-1 : -17 : 3)\) | \((-3 : 24 : 2)\) |
| Known points | |||||
|---|---|---|---|---|---|
| \((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
| \((2 : -17 : 1)\) | \((2 : 17 : 1)\) | \((-1 : -17 : 3)\) | \((-1 : 17 : 3)\) | \((-3 : -17 : 2)\) | \((-3 : 17 : 2)\) |
Number of rational Weierstrass points: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z\)
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.128091\) | \(\infty\) |
| \((-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 - z^3\) | \(0.128091\) | \(\infty\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.128091\) | \(\infty\) |
| \((-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-2xz^2 - z^3\) | \(0.128091\) | \(\infty\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 - z^3\) | \(0.128091\) | \(\infty\) |
| \((-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 - 3xz^2 - z^3\) | \(0.128091\) | \(\infty\) |
2-torsion field: 6.0.2728448.1
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(2\) |
| Mordell-Weil rank: | \(2\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 0.012305 \) |
| Real period: | \( 20.94165 \) |
| Tamagawa product: | \( 3 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 0.773095 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(2\) | \(3\) | \(3\) | \(1 + T + T^{2}\) | |
| \(73\) | \(2\) | \(2\) | \(1\) | \(1 + 17 T + 73 T^{2}\) |  |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
| Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
|---|---|---|
| \(2\) | 2.40.3 | no |
| \(3\) | 3.480.12 | no |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $E_6$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 6.6.2073071593.1 with defining polynomial:
\(x^{6} - x^{5} - 30 x^{4} + 31 x^{3} + 206 x^{2} - 150 x - 81\)
Decomposes up to isogeny as the square of the elliptic curve isogeny class:
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = -\frac{129991}{324} b^{5} - \frac{29090}{27} b^{4} + \frac{374003}{54} b^{3} + \frac{4820093}{324} b^{2} - \frac{1726571}{108} b - \frac{30747}{4}\)
\(g_6 = -\frac{563289097}{2916} b^{5} - \frac{150339193}{243} b^{4} + \frac{187103891}{54} b^{3} + \frac{23783771351}{2916} b^{2} - \frac{8144724245}{972} b - \frac{144295669}{36}\)
Conductor norm: 64
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\C\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.2073071593.1 with defining polynomial \(x^{6} - x^{5} - 30 x^{4} + 31 x^{3} + 206 x^{2} - 150 x - 81\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
| \(\End (J_{\overline{\Q}})\) | \(\simeq\) | an Eichler order of index \(3\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\) |
| \(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\mathrm{M}_2(\)\(\Q\)\()\) |
| \(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\mathrm{M}_2 (\R)\) |
Remainder of the endomorphism lattice by field
Over subfield \(F \simeq \) \(\Q(\sqrt{73}) \) with generator \(-\frac{1}{27} a^{5} - \frac{4}{27} a^{4} + \frac{19}{27} a^{3} + \frac{73}{27} a^{2} - \frac{22}{9} a - 5\) with minimal polynomial \(x^{2} - x - 18\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.5329.1 with generator \(\frac{1}{36} a^{5} + \frac{1}{9} a^{4} - \frac{5}{18} a^{3} - \frac{55}{36} a^{2} - \frac{35}{12} a + \frac{1}{4}\) with minimal polynomial \(x^{3} - x^{2} - 24 x + 27\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple