Minimal equation
Minimal equation
Simplified equation
| $y^2 = 2x^5 + 5x^3 + 3x$ | (homogenize, simplify) |
| $y^2 = 2x^5z + 5x^3z^3 + 3xz^5$ | (dehomogenize, simplify) |
| $y^2 = 2x^5 + 5x^3 + 3x$ | (homogenize, minimize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(147456\) | \(=\) | \( 2^{14} \cdot 3^{2} \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(147456,2),R![1]>*])); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(884736\) | \(=\) | \( 2^{15} \cdot 3^{3} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(195\) | \(=\) | \( 3 \cdot 5 \cdot 13 \) |
| \( I_4 \) | \(=\) | \(630\) | \(=\) | \( 2 \cdot 3^{2} \cdot 5 \cdot 7 \) |
| \( I_6 \) | \(=\) | \(44910\) | \(=\) | \( 2 \cdot 3^{2} \cdot 5 \cdot 499 \) |
| \( I_{10} \) | \(=\) | \(108\) | \(=\) | \( 2^{2} \cdot 3^{3} \) |
| \( J_2 \) | \(=\) | \(780\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 13 \) |
| \( J_4 \) | \(=\) | \(18630\) | \(=\) | \( 2 \cdot 3^{4} \cdot 5 \cdot 23 \) |
| \( J_6 \) | \(=\) | \(-380\) | \(=\) | \( - 2^{2} \cdot 5 \cdot 19 \) |
| \( J_8 \) | \(=\) | \(-86843325\) | \(=\) | \( - 3 \cdot 5^{2} \cdot 113 \cdot 10247 \) |
| \( J_{10} \) | \(=\) | \(884736\) | \(=\) | \( 2^{15} \cdot 3^{3} \) |
| \( g_1 \) | \(=\) | \(10442615625/32\) | ||
| \( g_2 \) | \(=\) | \(2558131875/256\) | ||
| \( g_3 \) | \(=\) | \(-401375/1536\) |
Automorphism group
| \(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
| \(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_4$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
Number of rational Weierstrass points: \(2\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(2x^2 + 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(2x^2 + 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(2x^2 + 3z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
2-torsion field: \(\Q(i, \sqrt{6})\)
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(0\) |
| Mordell-Weil rank: | \(0\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 1 \) |
| Real period: | \( 5.583154 \) |
| Tamagawa product: | \( 4 \) |
| Torsion order: | \( 4 \) |
| Leading coefficient: | \( 1.395788 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(14\) | \(15\) | \(2\) | \(1\) | |
| \(3\) | \(2\) | \(3\) | \(2\) | \(1 + 3 T^{2}\) |  |
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? |
|---|---|---|
| \(2\) | 2.180.3 | yes |
| \(3\) | 3.270.1 | no |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $J(E_4)$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) 4.2.55296.4 with defining polynomial:
\(x^{4} - 24\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = 240 b^{2} - 720\)
\(g_6 = 4320 b^{3} - 16128 b\)
Conductor norm: 32
\(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = 240 b^{2} - 720\)
\(g_6 = -4320 b^{3} + 16128 b\)
Conductor norm: 32
Endomorphisms of the Jacobian
Not of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
| \(\End (J_{})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 8.0.12230590464.5 with defining polynomial \(x^{8} + 4 x^{6} + 18 x^{4} - 68 x^{2} + 49\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
| \(\End (J_{\overline{\Q}})\) | \(\simeq\) | a non-Eichler order of index \(4\) 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 \(F \simeq \) \(\Q(\sqrt{-1}) \) with generator \(\frac{5}{56} a^{7} + \frac{27}{56} a^{5} + \frac{125}{56} a^{3} - \frac{193}{56} a\) with minimal polynomial \(x^{2} + 1\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{-1}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{-6}) \) with generator \(\frac{1}{4} a^{6} + \frac{5}{4} a^{4} + \frac{25}{4} a^{2} - \frac{39}{4}\) with minimal polynomial \(x^{2} + 6\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{6}) \) with generator \(\frac{1}{28} a^{7} + \frac{1}{7} a^{5} + \frac{11}{28} a^{3} - \frac{41}{14} a\) with minimal polynomial \(x^{2} - 6\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R\) |
Not of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(i, \sqrt{6})\) with generator \(-\frac{1}{56} a^{7} + \frac{1}{8} a^{6} - \frac{1}{14} a^{5} + \frac{5}{8} a^{4} - \frac{11}{56} a^{3} + \frac{25}{8} a^{2} + \frac{41}{28} a - \frac{39}{8}\) with minimal polynomial \(x^{4} + 9\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{-1}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-1}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 4.0.55296.1 with generator \(\frac{1}{8} a^{6} + \frac{5}{8} a^{4} + \frac{21}{8} a^{2} - \frac{43}{8}\) with minimal polynomial \(x^{4} + 6\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{2}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 4.0.55296.1 with generator \(\frac{5}{56} a^{7} + \frac{27}{56} a^{5} + \frac{125}{56} a^{3} - \frac{137}{56} a\) with minimal polynomial \(x^{4} + 6\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{2}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 4.2.55296.4 with generator \(-\frac{5}{56} a^{7} + \frac{1}{8} a^{6} - \frac{27}{56} a^{5} + \frac{5}{8} a^{4} - \frac{125}{56} a^{3} + \frac{21}{8} a^{2} + \frac{137}{56} a - \frac{43}{8}\) with minimal polynomial \(x^{4} - 24\):
| \(\End (J_{F})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, not simple
Over subfield \(F \simeq \) 4.2.55296.4 with generator \(\frac{5}{56} a^{7} + \frac{1}{8} a^{6} + \frac{27}{56} a^{5} + \frac{5}{8} a^{4} + \frac{125}{56} a^{3} + \frac{21}{8} a^{2} - \frac{137}{56} a - \frac{43}{8}\) with minimal polynomial \(x^{4} - 24\):
| \(\End (J_{F})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, not simple