Properties

Label 1050.a.131250.1
Conductor $1050$
Discriminant $-131250$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{4}\Z\)
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

$y^2 + (x^2 + x)y = 3x^6 + 8x^5 + 15x^4 + 17x^3 + 15x^2 + 8x + 3$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = 3x^6 + 8x^5z + 15x^4z^2 + 17x^3z^3 + 15x^2z^4 + 8xz^5 + 3z^6$ (dehomogenize, simplify)
$y^2 = 12x^6 + 32x^5 + 61x^4 + 70x^3 + 61x^2 + 32x + 12$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([3, 8, 15, 17, 15, 8, 3]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![3, 8, 15, 17, 15, 8, 3], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([12, 32, 61, 70, 61, 32, 12]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1050\) \(=\) \( 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-131250\) \(=\) \( - 2 \cdot 3 \cdot 5^{5} \cdot 7 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(11868\) \(=\)  \( 2^{2} \cdot 3 \cdot 23 \cdot 43 \)
\( I_4 \)  \(=\) \(198609\) \(=\)  \( 3 \cdot 239 \cdot 277 \)
\( I_6 \)  \(=\) \(759217863\) \(=\)  \( 3 \cdot 12107 \cdot 20903 \)
\( I_{10} \)  \(=\) \(16800000\) \(=\)  \( 2^{8} \cdot 3 \cdot 5^{5} \cdot 7 \)
\( J_2 \)  \(=\) \(2967\) \(=\)  \( 3 \cdot 23 \cdot 43 \)
\( J_4 \)  \(=\) \(358520\) \(=\)  \( 2^{3} \cdot 5 \cdot 8963 \)
\( J_6 \)  \(=\) \(56735700\) \(=\)  \( 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 27017 \)
\( J_8 \)  \(=\) \(9949557875\) \(=\)  \( 5^{3} \cdot 79596463 \)
\( J_{10} \)  \(=\) \(131250\) \(=\)  \( 2 \cdot 3 \cdot 5^{5} \cdot 7 \)
\( g_1 \)  \(=\) \(76641937806559869/43750\)
\( g_2 \)  \(=\) \(312136655012892/4375\)
\( g_3 \)  \(=\) \(475666111026/125\)

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 $\Q_{5}$.

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: \(\Z/{2}\Z \times \Z/{4}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 8.0.497871360000.3

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(3\)
Regulator: \( 1 \)
Real period: \( 6.612551 \)
Tamagawa product: \( 2 \)
Torsion order:\( 8 \)
Leading coefficient: \( 0.413284 \)
Analytic order of Ш: \( 2 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + T + 2 T^{2} )\)
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 3 T^{2} )\)
\(5\) \(2\) \(5\) \(2\) \(( 1 - T )( 1 + T )\)
\(7\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 7 T^{2} )\)

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 15.a
  Elliptic curve isogeny class 70.a

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 \(\Q\).