Properties

Label 12544.h.401408.1
Conductor $12544$
Discriminant $401408$
Mordell-Weil group \(\Z \times \Z/{2}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 = x^5 - x^4 + 4x^3 - x^2 + x$ (homogenize, simplify)
$y^2 = x^5z - x^4z^2 + 4x^3z^3 - x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = x^5 - x^4 + 4x^3 - x^2 + x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -1, 4, -1, 1]), R([]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -1, 4, -1, 1], R![]);
 
sage: X = HyperellipticCurve(R([0, 1, -1, 4, -1, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(12544\) \(=\) \( 2^{8} \cdot 7^{2} \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(12544,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(401408\) \(=\) \( 2^{13} \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(120\) \(=\)  \( 2^{3} \cdot 3 \cdot 5 \)
\( I_4 \)  \(=\) \(408\) \(=\)  \( 2^{3} \cdot 3 \cdot 17 \)
\( I_6 \)  \(=\) \(14988\) \(=\)  \( 2^{2} \cdot 3 \cdot 1249 \)
\( I_{10} \)  \(=\) \(1568\) \(=\)  \( 2^{5} \cdot 7^{2} \)
\( J_2 \)  \(=\) \(240\) \(=\)  \( 2^{4} \cdot 3 \cdot 5 \)
\( J_4 \)  \(=\) \(1312\) \(=\)  \( 2^{5} \cdot 41 \)
\( J_6 \)  \(=\) \(-2048\) \(=\)  \( - 2^{11} \)
\( J_8 \)  \(=\) \(-553216\) \(=\)  \( - 2^{8} \cdot 2161 \)
\( J_{10} \)  \(=\) \(401408\) \(=\)  \( 2^{13} \cdot 7^{2} \)
\( g_1 \)  \(=\) \(97200000/49\)
\( g_2 \)  \(=\) \(2214000/49\)
\( g_3 \)  \(=\) \(-14400/49\)

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : -2 : 1),\, (1 : 2 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : -2 : 1),\, (1 : 2 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : -1 : 1),\, (1 : 1 : 1)\)

magma: [C![0,0,1],C![1,-2,1],C![1,0,0],C![1,2,1]]; // minimal model
 
magma: [C![0,0,1],C![1,-1,1],C![1,0,0],C![1,1,1]]; // simplified model
 

Number of rational Weierstrass points: \(2\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable everywhere.

magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
 

Mordell-Weil group of the Jacobian

Group structure: \(\Z \times \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.119959\) \(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((1 : -2 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.119959\) \(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.119959\) \(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)

2-torsion field: 4.0.392.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.119959 \)
Real period: \( 7.745608 \)
Tamagawa product: \( 4 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.929162 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(8\) \(13\) \(4\) \(1\)
\(7\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

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 112.a
  Elliptic curve isogeny class 112.b

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).