Minimal equation
Minimal equation
Simplified equation
| $y^2 + (x^3 + x + 1)y = -3x^4 - 3x^3 + x^2 + x$ | (homogenize, simplify) |
| $y^2 + (x^3 + xz^2 + z^3)y = -3x^4z^2 - 3x^3z^3 + x^2z^4 + xz^5$ | (dehomogenize, simplify) |
| $y^2 = x^6 - 10x^4 - 10x^3 + 5x^2 + 6x + 1$ | (minimize, homogenize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(17689\) | \(=\) | \( 7^{2} \cdot 19^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(17689\) | \(=\) | \( 7^{2} \cdot 19^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(580\) | \(=\) | \( 2^{2} \cdot 5 \cdot 29 \) |
| \( I_4 \) | \(=\) | \(11305\) | \(=\) | \( 5 \cdot 7 \cdot 17 \cdot 19 \) |
| \( I_6 \) | \(=\) | \(1902565\) | \(=\) | \( 5 \cdot 7 \cdot 19 \cdot 2861 \) |
| \( I_{10} \) | \(=\) | \(2264192\) | \(=\) | \( 2^{7} \cdot 7^{2} \cdot 19^{2} \) |
| \( J_2 \) | \(=\) | \(145\) | \(=\) | \( 5 \cdot 29 \) |
| \( J_4 \) | \(=\) | \(405\) | \(=\) | \( 3^{4} \cdot 5 \) |
| \( J_6 \) | \(=\) | \(-395\) | \(=\) | \( - 5 \cdot 79 \) |
| \( J_8 \) | \(=\) | \(-55325\) | \(=\) | \( - 5^{2} \cdot 2213 \) |
| \( J_{10} \) | \(=\) | \(17689\) | \(=\) | \( 7^{2} \cdot 19^{2} \) |
| \( g_1 \) | \(=\) | \(64097340625/17689\) | ||
| \( g_2 \) | \(=\) | \(1234693125/17689\) | ||
| \( g_3 \) | \(=\) | \(-8304875/17689\) |
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 | |||||
|---|---|---|---|---|---|
| \((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : 1 : 1)\) |
| \((5 : -20 : 1)\) | \((-1 : -44 : 6)\) | \((-6 : 75 : 5)\) | \((5 : -111 : 1)\) | \((-1 : -135 : 6)\) | \((-6 : 166 : 5)\) |
Number of rational Weierstrass points: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \times \Z\)
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.166459\) | \(\infty\) |
| \((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.166459\) | \(\infty\) |
2-torsion field: 9.9.5534900853769.1
BSD invariants
| Hasse-Weil conjecture: | verified |
| Analytic rank: | \(2\) |
| Mordell-Weil rank: | \(2\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 0.020781 \) |
| Real period: | \( 23.05984 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 1 \) |
| Leading coefficient: | \( 0.479221 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(7\) | \(2\) | \(2\) | \(1\) | \(1 + 4 T + 7 T^{2}\) |  |
| \(19\) | \(2\) | \(2\) | \(1\) | \(1 - T + 19 T^{2}\) | |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $E_6$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 6.6.41615795893.2 with defining polynomial:
\(x^{6} - x^{5} - 55 x^{4} + 160 x^{3} + 246 x^{2} - 1107 x + 729\)
Decomposes up to isogeny as the square of the elliptic curve:
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = -\frac{29571725}{89424} b^{5} + \frac{12875}{552} b^{4} + \frac{1628558075}{89424} b^{3} - \frac{268027405}{7452} b^{2} - \frac{95055545}{828} b + \frac{5963540}{23}\)
\(g_6 = \frac{391613779963}{3219264} b^{5} - \frac{1529536043}{178848} b^{4} - \frac{21563971628059}{3219264} b^{3} + \frac{7102017353893}{536544} b^{2} + \frac{2517427711897}{59616} b - \frac{421325658103}{4416}\)
Conductor norm: 1
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 \(K = \Q (a) \simeq \) 6.6.41615795893.2 with defining polynomial \(x^{6} - x^{5} - 55 x^{4} + 160 x^{3} + 246 x^{2} - 1107 x + 729\)
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 \(F \simeq \) \(\Q(\sqrt{133}) \) with generator \(\frac{4}{621} a^{5} + \frac{5}{621} a^{4} - \frac{157}{621} a^{3} + \frac{442}{621} a^{2} - \frac{134}{207} a - \frac{117}{23}\) with minimal polynomial \(x^{2} - x - 33\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.17689.1 with generator \(-\frac{16}{207} a^{5} - \frac{20}{207} a^{4} + \frac{835}{207} a^{3} - \frac{733}{207} a^{2} - \frac{1879}{69} a + \frac{760}{23}\) with minimal polynomial \(x^{3} - x^{2} - 44 x - 69\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple