Properties

Label 1077.a.1077.1
Conductor $1077$
Discriminant $1077$
Mordell-Weil group \(\Z \times \Z/{2}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = 5x^5 + 34x^4 + 80x^3 - x^2 - 90x + 32$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = 5x^5z + 34x^4z^2 + 80x^3z^3 - x^2z^4 - 90xz^5 + 32z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 20x^5 + 136x^4 + 322x^3 - 4x^2 - 360x + 129$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([32, -90, -1, 80, 34, 5]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![32, -90, -1, 80, 34, 5], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([129, -360, -4, 322, 136, 20, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1077\) \(=\) \( 3 \cdot 359 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(1077\) \(=\) \( 3 \cdot 359 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(155924\) \(=\)  \( 2^{2} \cdot 17 \cdot 2293 \)
\( I_4 \)  \(=\) \(161593\) \(=\)  \( 283 \cdot 571 \)
\( I_6 \)  \(=\) \(8379938029\) \(=\)  \( 8379938029 \)
\( I_{10} \)  \(=\) \(137856\) \(=\)  \( 2^{7} \cdot 3 \cdot 359 \)
\( J_2 \)  \(=\) \(38981\) \(=\)  \( 17 \cdot 2293 \)
\( J_4 \)  \(=\) \(63306532\) \(=\)  \( 2^{2} \cdot 61 \cdot 259453 \)
\( J_6 \)  \(=\) \(137068427976\) \(=\)  \( 2^{3} \cdot 3 \cdot 5711184499 \)
\( J_8 \)  \(=\) \(333836849266358\) \(=\)  \( 2 \cdot 20369 \cdot 8194728491 \)
\( J_{10} \)  \(=\) \(1077\) \(=\)  \( 3 \cdot 359 \)
\( g_1 \)  \(=\) \(90004636142290020118901/1077\)
\( g_2 \)  \(=\) \(3749794358746968581012/1077\)
\( g_3 \)  \(=\) \(69425997674312689112/359\)

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

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-13 : 1094 : 2),\, (-13 : 1095 : 2)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-13 : 1094 : 2),\, (-13 : 1095 : 2)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-13 : -1 : 2),\, (-13 : 1 : 2)\)

magma: [C![-13,1094,2],C![-13,1095,2],C![1,-1,0],C![1,0,0]]; // minimal model
 
magma: [C![-13,-1,2],C![-13,1,2],C![1,-1,0],C![1,1,0]]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 6xz - 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(-20xz^2 + 11z^3\) \(0.035632\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 6xz - 3z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-39xz^2 + 17z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 6xz - 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(-20xz^2 + 11z^3\) \(0.035632\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 6xz - 3z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-39xz^2 + 17z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + 6xz - 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 40xz^2 + 23z^3\) \(0.035632\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + 6xz - 3z^2\) \(=\) \(0,\) \(2y\) \(=\) \(x^3 - 78xz^2 + 35z^3\) \(0\) \(2\)

2-torsion field: 6.6.890825472.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.035632 \)
Real period: \( 21.23503 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.189165 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(359\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 8 T + 359 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

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