Minimal equation
Minimal equation
Simplified equation
| $y^2 = x^5 - 2x^3 - x$ | (homogenize, simplify) |
| $y^2 = x^5z - 2x^3z^3 - xz^5$ | (dehomogenize, simplify) |
| $y^2 = x^5 - 2x^3 - x$ | (homogenize, minimize) |
Invariants
| Conductor: | \( N \) | \(=\) | \(262144\) | \(=\) | \( 2^{18} \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(262144,2),R![1]>*])); Factorization($1);
|
| Discriminant: | \( \Delta \) | \(=\) | \(-262144\) | \(=\) | \( - 2^{18} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
| \( I_2 \) | \(=\) | \(4\) | \(=\) | \( 2^{2} \) |
| \( I_4 \) | \(=\) | \(-14\) | \(=\) | \( - 2 \cdot 7 \) |
| \( I_6 \) | \(=\) | \(2\) | \(=\) | \( 2 \) |
| \( I_{10} \) | \(=\) | \(1\) | \(=\) | \( 1 \) |
| \( J_2 \) | \(=\) | \(32\) | \(=\) | \( 2^{5} \) |
| \( J_4 \) | \(=\) | \(640\) | \(=\) | \( 2^{7} \cdot 5 \) |
| \( J_6 \) | \(=\) | \(-6144\) | \(=\) | \( - 2^{11} \cdot 3 \) |
| \( J_8 \) | \(=\) | \(-151552\) | \(=\) | \( - 2^{12} \cdot 37 \) |
| \( J_{10} \) | \(=\) | \(262144\) | \(=\) | \( 2^{18} \) |
| \( g_1 \) | \(=\) | \(128\) | ||
| \( g_2 \) | \(=\) | \(80\) | ||
| \( g_3 \) | \(=\) | \(-24\) |
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: \(\Z \oplus \Z/{2}\Z\)
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(1.123512\) | \(\infty\) |
| \((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(xz^2 + z^3\) | \(1.123512\) | \(\infty\) |
| \((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
| Generator | $D_0$ | Height | Order | |||||
|---|---|---|---|---|---|---|---|---|
| \(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(1/2xz^2 + 1/2z^3\) | \(1.123512\) | \(\infty\) |
| \((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
BSD invariants
| Hasse-Weil conjecture: | unverified |
| Analytic rank: | \(1\) |
| Mordell-Weil rank: | \(1\) |
| 2-Selmer rank: | \(2\) |
| Regulator: | \( 1.123512 \) |
| Real period: | \( 10.88380 \) |
| Tamagawa product: | \( 1 \) |
| Torsion order: | \( 2 \) |
| Leading coefficient: | \( 3.057022 \) |
| Analytic order of Ш: | \( 1 \) (rounded) |
| Order of Ш: | square |
Local invariants
| Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
|---|---|---|---|---|---|
| \(2\) | \(18\) | \(18\) | \(1\) | \(1\) |
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? |
|---|---|---|
| \(2\) | 2.90.3 | yes |
| \(3\) | 3.3240.5 | no |
Sato-Tate group
| \(\mathrm{ST}\) | \(\simeq\) | $J(E_2)$ |
| \(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{8})\) with defining polynomial:
\(x^{4} + 1\)
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
\(g_4 = -\frac{120320}{28561} b^{2} + \frac{489984}{28561}\)
\(g_6 = -\frac{27344896}{371293} b^{3} + \frac{15777792}{371293} b\)
Conductor norm: 1048576
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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{8})\) with defining polynomial \(x^{4} + 1\)
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 \(F \simeq \) \(\Q(\sqrt{-2}) \) with generator \(-a^{3} - a\) with minimal polynomial \(x^{2} + 2\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{2}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{2}) \) with generator \(a^{3} - a\) with minimal polynomial \(x^{2} - 2\):
| \(\End (J_{F})\) | \(\simeq\) | \(\Z [\sqrt{2}]\) |
| \(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{2}) \) |
| \(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) \(\Q(\sqrt{-1}) \) with generator \(-a^{2}\) 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