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Properties

Label 1573.b.224939.1
Conductor $1573$
Discriminant $-224939$
Mordell-Weil group \(\Z/{2}\Z \oplus \Z/{4}\Z\)
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

$y^2 + (x + 1)y = x^5 + x^4 - 5x^3 + 3x^2 - 1$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = x^5z + x^4z^2 - 5x^3z^3 + 3x^2z^4 - z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 + 4x^4 - 20x^3 + 13x^2 + 2x - 3$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 0, 3, -5, 1, 1]), R([1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 0, 3, -5, 1, 1], R![1, 1]);
 
sage: X = HyperellipticCurve(R([-3, 2, 13, -20, 4, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1573\) \(=\) \( 11^{2} \cdot 13 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-224939\) \(=\) \( - 11^{3} \cdot 13^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(472\) \(=\)  \( 2^{3} \cdot 59 \)
\( I_4 \)  \(=\) \(-4796\) \(=\)  \( - 2^{2} \cdot 11 \cdot 109 \)
\( I_6 \)  \(=\) \(-683705\) \(=\)  \( - 5 \cdot 11 \cdot 31 \cdot 401 \)
\( I_{10} \)  \(=\) \(-899756\) \(=\)  \( - 2^{2} \cdot 11^{3} \cdot 13^{2} \)
\( J_2 \)  \(=\) \(236\) \(=\)  \( 2^{2} \cdot 59 \)
\( J_4 \)  \(=\) \(3120\) \(=\)  \( 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
\( J_6 \)  \(=\) \(53993\) \(=\)  \( 53993 \)
\( J_8 \)  \(=\) \(751987\) \(=\)  \( 751987 \)
\( J_{10} \)  \(=\) \(-224939\) \(=\)  \( - 11^{3} \cdot 13^{2} \)
\( g_1 \)  \(=\) \(-732082482176/224939\)
\( g_2 \)  \(=\) \(-3154621440/17303\)
\( g_3 \)  \(=\) \(-3007194128/224939\)

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

magma: [C![-3,1,1],C![1,-1,1],C![1,0,0]]; // minimal model
 
magma: [C![-3,0,1],C![1,0,1],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: \(\Z/{2}\Z \oplus \Z/{4}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-3 : 1 : 1) - (1 : 0 : 0)\) \(x + 3z\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0\) \(4\)
Generator $D_0$ Height Order
\((-3 : 1 : 1) - (1 : 0 : 0)\) \(x + 3z\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0\) \(4\)
Generator $D_0$ Height Order
\((-3 : 0 : 1) - (1 : 0 : 0)\) \(x + 3z\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + 3z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + 3z^3\) \(0\) \(4\)

2-torsion field: 3.1.44.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 9.742030 \)
Tamagawa product: \( 4 \)
Torsion order:\( 8 \)
Leading coefficient: \( 0.608876 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(11\) \(2\) \(3\) \(2\) \(1 - 4 T + 11 T^{2}\)
\(13\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + 2 T + 13 T^{2} )\)

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

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);