Minimal equation
Minimal equation
Simplified equation
$y^2 = 2x^5 - x^4 - 5x^3 + 3x + 1$ | (homogenize, simplify) |
$y^2 = 2x^5z - x^4z^2 - 5x^3z^3 + 3xz^5 + z^6$ | (dehomogenize, simplify) |
$y^2 = 2x^5 - x^4 - 5x^3 + 3x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(15360\) | \(=\) | \( 2^{10} \cdot 3 \cdot 5 \) | magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(15360,2),R![1]>*])); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(184320\) | \(=\) | \( 2^{12} \cdot 3^{2} \cdot 5 \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(390\) | \(=\) | \( 2 \cdot 3 \cdot 5 \cdot 13 \) |
\( I_4 \) | \(=\) | \(852\) | \(=\) | \( 2^{2} \cdot 3 \cdot 71 \) |
\( I_6 \) | \(=\) | \(105720\) | \(=\) | \( 2^{3} \cdot 3 \cdot 5 \cdot 881 \) |
\( I_{10} \) | \(=\) | \(720\) | \(=\) | \( 2^{4} \cdot 3^{2} \cdot 5 \) |
\( J_2 \) | \(=\) | \(780\) | \(=\) | \( 2^{2} \cdot 3 \cdot 5 \cdot 13 \) |
\( J_4 \) | \(=\) | \(23078\) | \(=\) | \( 2 \cdot 11 \cdot 1049 \) |
\( J_6 \) | \(=\) | \(838980\) | \(=\) | \( 2^{2} \cdot 3^{2} \cdot 5 \cdot 59 \cdot 79 \) |
\( J_8 \) | \(=\) | \(30452579\) | \(=\) | \( 30452579 \) |
\( J_{10} \) | \(=\) | \(184320\) | \(=\) | \( 2^{12} \cdot 3^{2} \cdot 5 \) |
\( g_1 \) | \(=\) | \(6265569375/4\) | ||
\( g_2 \) | \(=\) | \(1901338725/32\) | ||
\( g_3 \) | \(=\) | \(177234525/64\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | 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: \(4\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.516278\) | \(\infty\) |
\((1 : 0 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 0 : 1) + (-1 : 0 : 2) - 2 \cdot(1 : 0 : 0)\) | \((x + z) (2x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.516278\) | \(\infty\) |
\((1 : 0 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 0 : 1) + (-1 : 0 : 2) - 2 \cdot(1 : 0 : 0)\) | \((x + z) (2x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : -1/2 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-1/2z^3\) | \(0.516278\) | \(\infty\) |
\((1 : 0 : 1) - (1 : 0 : 0)\) | \(x - z\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \((x - z) (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((-1 : 0 : 1) + (-1 : 0 : 2) - 2 \cdot(1 : 0 : 0)\) | \((x + z) (2x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
2-torsion field: \(\Q(\sqrt{5}) \)
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(1\) |
Mordell-Weil rank: | \(1\) |
2-Selmer rank: | \(4\) |
Regulator: | \( 0.516278 \) |
Real period: | \( 17.10135 \) |
Tamagawa product: | \( 8 \) |
Torsion order: | \( 8 \) |
Leading coefficient: | \( 1.103632 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(2\) | \(10\) | \(12\) | \(4\) | \(1\) | |
\(3\) | \(1\) | \(2\) | \(2\) | \(( 1 + T )( 1 + 3 T^{2} )\) | |
\(5\) | \(1\) | \(1\) | \(1\) | \(( 1 - T )( 1 + 2 T + 5 T^{2} )\) |
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? |
---|---|---|
\(2\) | 2.360.2 | yes |
\(3\) | 3.270.2 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $N(\mathrm{U}(1)\times\mathrm{SU}(2))$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{U}(1)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 480.c
Elliptic curve isogeny class 32.a
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)
Not of \(\GL_2\)-type over \(\overline{\Q}\)
Endomorphism ring over \(\overline{\Q}\):
\(\End (J_{\overline{\Q}})\) | \(\simeq\) | an order of index \(4\) in \(\Z \times \Z [\sqrt{-1}]\) |
\(\End (J_{\overline{\Q}}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q(\sqrt{-1}) \) |
\(\End (J_{\overline{\Q}}) \otimes \R\) | \(\simeq\) | \(\R \times \C\) |