Properties

Label 30912.d.92736.1
Conductor $30912$
Discriminant $-92736$
Mordell-Weil group \(\Z/{6}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + 1)y = -7x^6 - 42x^4 - 82x^2 - 52$ (homogenize, simplify)
$y^2 + (x^2z + z^3)y = -7x^6 - 42x^4z^2 - 82x^2z^4 - 52z^6$ (dehomogenize, simplify)
$y^2 = -28x^6 - 167x^4 - 326x^2 - 207$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-52, 0, -82, 0, -42, 0, -7]), R([1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-52, 0, -82, 0, -42, 0, -7], R![1, 0, 1]);
 
sage: X = HyperellipticCurve(R([-207, 0, -326, 0, -167, 0, -28]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(30912\) \(=\) \( 2^{6} \cdot 3 \cdot 7 \cdot 23 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-92736\) \(=\) \( - 2^{6} \cdot 3^{2} \cdot 7 \cdot 23 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(282764\) \(=\)  \( 2^{2} \cdot 223 \cdot 317 \)
\( I_4 \)  \(=\) \(1297315\) \(=\)  \( 5 \cdot 23 \cdot 29 \cdot 389 \)
\( I_6 \)  \(=\) \(121785864076\) \(=\)  \( 2^{2} \cdot 53 \cdot 347 \cdot 1655509 \)
\( I_{10} \)  \(=\) \(11592\) \(=\)  \( 2^{3} \cdot 3^{2} \cdot 7 \cdot 23 \)
\( J_2 \)  \(=\) \(282764\) \(=\)  \( 2^{2} \cdot 223 \cdot 317 \)
\( J_4 \)  \(=\) \(3330613444\) \(=\)  \( 2^{2} \cdot 2579 \cdot 322859 \)
\( J_6 \)  \(=\) \(52294241220480\) \(=\)  \( 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23 \cdot 56390443 \)
\( J_8 \)  \(=\) \(923485727778566396\) \(=\)  \( 2^{2} \cdot 53 \cdot 313 \cdot 13917139788091 \)
\( J_{10} \)  \(=\) \(92736\) \(=\)  \( 2^{6} \cdot 3^{2} \cdot 7 \cdot 23 \)
\( g_1 \)  \(=\) \(28244936909000890514471216/1449\)
\( g_2 \)  \(=\) \(1176566846346573015943724/1449\)
\( g_3 \)  \(=\) \(45087249203349453280\)

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

This curve has no rational points.
This curve has no rational points.
This curve has no rational points.

magma: []; // minimal model
 
magma: []; // simplified model
 

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable except over $\R$.

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(x^2 + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^2z + 3z^3\) \(0\) \(6\)

2-torsion field: 8.0.1698758656.6

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(4\)
Regulator: \( 1 \)
Real period: \( 2.622996 \)
Tamagawa product: \( 2 \)
Torsion order:\( 6 \)
Leading coefficient: \( 1.165776 \)
Analytic order of Ш: \( 8 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(6\) \(6\) \(1\) \(1 + T\)
\(3\) \(1\) \(2\) \(2\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(7\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 4 T + 7 T^{2} )\)
\(23\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 23 T^{2} )\)

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?
\(2\) 2.45.1 yes
\(3\) 3.720.4 yes

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 2208.j
  Elliptic curve isogeny class 14.a

magma: HeuristicDecompositionFactors(C);
 

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 \(\Q\).

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);