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Properties

Label 1272.a.122112.1
Conductor $1272$
Discriminant $-122112$
Mordell-Weil group \(\Z/{2}\Z \oplus \Z/{6}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + 1)y = 3x^5 + 4x^4 + 2x^3 - x^2 - x$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = 3x^5z + 4x^4z^2 + 2x^3z^3 - x^2z^4 - xz^5$ (dehomogenize, simplify)
$y^2 = 12x^5 + 17x^4 + 8x^3 - 2x^2 - 4x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -1, 2, 4, 3]), R([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -1, 2, 4, 3], R![1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, -4, -2, 8, 17, 12]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1272\) \(=\) \( 2^{3} \cdot 3 \cdot 53 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-122112\) \(=\) \( - 2^{8} \cdot 3^{2} \cdot 53 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(124\) \(=\)  \( 2^{2} \cdot 31 \)
\( I_4 \)  \(=\) \(-5027\) \(=\)  \( - 11 \cdot 457 \)
\( I_6 \)  \(=\) \(-35457\) \(=\)  \( - 3 \cdot 53 \cdot 223 \)
\( I_{10} \)  \(=\) \(15264\) \(=\)  \( 2^{5} \cdot 3^{2} \cdot 53 \)
\( J_2 \)  \(=\) \(124\) \(=\)  \( 2^{2} \cdot 31 \)
\( J_4 \)  \(=\) \(3992\) \(=\)  \( 2^{3} \cdot 499 \)
\( J_6 \)  \(=\) \(-79504\) \(=\)  \( - 2^{4} \cdot 4969 \)
\( J_8 \)  \(=\) \(-6448640\) \(=\)  \( - 2^{9} \cdot 5 \cdot 11 \cdot 229 \)
\( J_{10} \)  \(=\) \(122112\) \(=\)  \( 2^{8} \cdot 3^{2} \cdot 53 \)
\( g_1 \)  \(=\) \(114516604/477\)
\( g_2 \)  \(=\) \(29731418/477\)
\( g_3 \)  \(=\) \(-4775209/477\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -1 : 1),\, (1 : -15 : 3)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -1 : 1),\, (1 : -15 : 3)\)
All points: \((1 : 0 : 0),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (1 : 0 : 3)\)

magma: [C![-1,-1,1],C![0,-1,1],C![0,0,1],C![1,-15,3],C![1,0,0]]; // minimal model
 
magma: [C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,0,3],C![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(3\)

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: \(\Z/{2}\Z \oplus \Z/{6}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
\((-1 : -1 : 1) + (0 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0\) \(6\)
Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
\((-1 : -1 : 1) + (0 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0\) \(6\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - z^3\) \(0\) \(2\)
\((-1 : 0 : 1) + (0 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^2z + 2xz^2 + z^3\) \(0\) \(6\)

2-torsion field: 3.1.212.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 14.39791 \)
Tamagawa product: \( 4 \)
Torsion order:\( 12 \)
Leading coefficient: \( 0.399942 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(3\) \(8\) \(2\) \(1 + T\)
\(3\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 - T + 3 T^{2} )\)
\(53\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 6 T + 53 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.120.3 yes
\(3\) 3.80.1 yes

Sato-Tate group

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

Decomposition of the Jacobian

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

magma: HeuristicDecompositionFactors(C);
 

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 \(\Q\).

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);