Minimal equation
Minimal equation
Simplified equation
$y^2 = x^5 + 3x^4 + 6x^3 + 6x^2 + 4x$ | (homogenize, simplify) |
$y^2 = x^5z + 3x^4z^2 + 6x^3z^3 + 6x^2z^4 + 4xz^5$ | (dehomogenize, simplify) |
$y^2 = x^5 + 3x^4 + 6x^3 + 6x^2 + 4x$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(28672\) | \(=\) | \( 2^{12} \cdot 7 \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(28672,2),R![1]>*])); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(458752\) | \(=\) | \( 2^{16} \cdot 7 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(44\) | \(=\) | \( 2^{2} \cdot 11 \) |
\( I_4 \) | \(=\) | \(76\) | \(=\) | \( 2^{2} \cdot 19 \) |
\( I_6 \) | \(=\) | \(1832\) | \(=\) | \( 2^{3} \cdot 229 \) |
\( I_{10} \) | \(=\) | \(56\) | \(=\) | \( 2^{3} \cdot 7 \) |
\( J_2 \) | \(=\) | \(176\) | \(=\) | \( 2^{4} \cdot 11 \) |
\( J_4 \) | \(=\) | \(480\) | \(=\) | \( 2^{5} \cdot 3 \cdot 5 \) |
\( J_6 \) | \(=\) | \(-51968\) | \(=\) | \( - 2^{8} \cdot 7 \cdot 29 \) |
\( J_8 \) | \(=\) | \(-2344192\) | \(=\) | \( - 2^{8} \cdot 9157 \) |
\( J_{10} \) | \(=\) | \(458752\) | \(=\) | \( 2^{16} \cdot 7 \) |
\( g_1 \) | \(=\) | \(2576816/7\) | ||
\( g_2 \) | \(=\) | \(39930/7\) | ||
\( g_3 \) | \(=\) | \(-3509\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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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 | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 2xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 2xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\(D_0 - 2 \cdot(1 : 0 : 0)\) | \(x^2 + 2xz + 2z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
2-torsion field: \(\Q(i, \sqrt{7})\)
BSD invariants
Hasse-Weil conjecture: | unverified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 5.751196 \) |
Tamagawa product: | \( 4 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 1.437799 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(12\) | \(16\) | \(4\) | \(1\) | |
\(7\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 - 4 T + 7 T^{2} )\) |
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 |
\(3\) | 3.45.1 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial:
\(x^{2} - 2\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 2.2.8.1-448.2-a
Elliptic curve isogeny class 2.2.8.1-448.1-a
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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial \(x^{2} - 2\)
Of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |