Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^2 + x)y = x^5 - 9x^4 + 12x^3 - 4x^2 - x$ | (homogenize, simplify) |
$y^2 + (x^2z + xz^2)y = x^5z - 9x^4z^2 + 12x^3z^3 - 4x^2z^4 - xz^5$ | (dehomogenize, simplify) |
$y^2 = 4x^5 - 35x^4 + 50x^3 - 15x^2 - 4x$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(537289\) | \(=\) | \( 733^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(537289\) | \(=\) | \( 733^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(2980\) | \(=\) | \( 2^{2} \cdot 5 \cdot 149 \) |
\( I_4 \) | \(=\) | \(502105\) | \(=\) | \( 5 \cdot 137 \cdot 733 \) |
\( I_6 \) | \(=\) | \(380797165\) | \(=\) | \( 5 \cdot 7 \cdot 733 \cdot 14843 \) |
\( I_{10} \) | \(=\) | \(68772992\) | \(=\) | \( 2^{7} \cdot 733^{2} \) |
\( J_2 \) | \(=\) | \(745\) | \(=\) | \( 5 \cdot 149 \) |
\( J_4 \) | \(=\) | \(2205\) | \(=\) | \( 3^{2} \cdot 5 \cdot 7^{2} \) |
\( J_6 \) | \(=\) | \(-2195\) | \(=\) | \( - 5 \cdot 439 \) |
\( J_8 \) | \(=\) | \(-1624325\) | \(=\) | \( - 5^{2} \cdot 43 \cdot 1511 \) |
\( J_{10} \) | \(=\) | \(537289\) | \(=\) | \( 733^{2} \) |
\( g_1 \) | \(=\) | \(229499299215625/537289\) | ||
\( g_2 \) | \(=\) | \(911753443125/537289\) | ||
\( g_3 \) | \(=\) | \(-1218279875/537289\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_6$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $D_6$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
Number of rational Weierstrass points: \(3\)
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 | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((0 : 0 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0\) | \(2\) |
\((0 : 0 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-xz^2\) | \(0\) | \(2\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((0 : 0 : 1) - (1 : 0 : 0)\) | \(x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z + xz^2\) | \(0\) | \(2\) |
\((0 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) | \(x (x - z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^2z - xz^2\) | \(0\) | \(2\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(0\) |
Mordell-Weil rank: | \(0\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 1 \) |
Real period: | \( 5.934576 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 4 \) |
Leading coefficient: | \( 3.338199 \) |
Analytic order of Ш: | \( 9 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(733\) | \(2\) | \(2\) | \(1\) | \(1 + 50 T + 733 T^{2}\) |
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? |
---|---|---|
\(2\) | 2.240.1 | yes |
\(3\) | 3.480.12 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $E_6$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Splits over the number field \(\Q (b)\) with defining polynomial:
\(x^{6} - x^{5} - 305 x^{4} - 3190 x^{3} - 13345 x^{2} - 24521 x - 15791\)
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
\(g_4 = -\frac{8658775}{12288} b^{5} + \frac{3452205}{2048} b^{4} + \frac{870793155}{4096} b^{3} + \frac{23982982945}{12288} b^{2} + \frac{41041673755}{6144} b + \frac{32513544835}{4096}\)
\(g_6 = -\frac{5246678395}{36864} b^{5} + \frac{50475704531}{147456} b^{4} + \frac{2110058027021}{49152} b^{3} + \frac{907064455213}{2304} b^{2} + \frac{66151883495221}{49152} b + \frac{235489792062851}{147456}\)
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 \(K = \Q (a)\) with defining polynomial \(x^{6} - x^{5} - 305 x^{4} - 3190 x^{3} - 13345 x^{2} - 24521 x - 15791\)
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 \(F \simeq \) \(\Q(\sqrt{733}) \) with generator \(\frac{1}{8} a^{5} - \frac{5}{8} a^{4} - \frac{71}{2} a^{3} - \frac{2063}{8} a^{2} - \frac{5333}{8} a - \frac{1073}{2}\) with minimal polynomial \(x^{2} - x - 183\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple
Over subfield \(F \simeq \) 3.3.537289.1 with generator \(-\frac{1}{9} a^{5} + \frac{5}{9} a^{4} + \frac{95}{3} a^{3} + \frac{2050}{9} a^{2} + \frac{1718}{3} a + \frac{3905}{9}\) with minimal polynomial \(x^{3} - x^{2} - 244 x - 1276\):
\(\End (J_{F})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{-3}}{2}]\) |
\(\End (J_{F}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{-3}) \) |
\(\End (J_{F}) \otimes \R\) | \(\simeq\) | \(\C\) |
Of \(\GL_2\)-type, simple