Properties

Label 630.a.34020.1
Conductor $630$
Discriminant $34020$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{2}\Z \times \Z/{4}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = 3x^5 + 10x^4 - 23x^2 - 6x + 15$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = 3x^5z + 10x^4z^2 - 23x^2z^4 - 6xz^5 + 15z^6$ (dehomogenize, simplify)
$y^2 = 12x^5 + 41x^4 + 2x^3 - 91x^2 - 24x + 60$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([15, -6, -23, 0, 10, 3]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![15, -6, -23, 0, 10, 3], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([60, -24, -91, 2, 41, 12]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(630\) \(=\) \( 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(34020\) \(=\) \( 2^{2} \cdot 3^{5} \cdot 5 \cdot 7 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(24100\) \(=\)  \( 2^{2} \cdot 5^{2} \cdot 241 \)
\( I_4 \)  \(=\) \(969793\) \(=\)  \( 11 \cdot 131 \cdot 673 \)
\( I_6 \)  \(=\) \(7474503265\) \(=\)  \( 5 \cdot 1494900653 \)
\( I_{10} \)  \(=\) \(4354560\) \(=\)  \( 2^{9} \cdot 3^{5} \cdot 5 \cdot 7 \)
\( J_2 \)  \(=\) \(6025\) \(=\)  \( 5^{2} \cdot 241 \)
\( J_4 \)  \(=\) \(1472118\) \(=\)  \( 2 \cdot 3 \cdot 73 \cdot 3361 \)
\( J_6 \)  \(=\) \(470090880\) \(=\)  \( 2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 89 \cdot 131 \)
\( J_8 \)  \(=\) \(166291536519\) \(=\)  \( 3^{3} \cdot 331 \cdot 1499 \cdot 12413 \)
\( J_{10} \)  \(=\) \(34020\) \(=\)  \( 2^{2} \cdot 3^{5} \cdot 5 \cdot 7 \)
\( g_1 \)  \(=\) \(1587871127345703125/6804\)
\( g_2 \)  \(=\) \(10732293030978125/1134\)
\( g_3 \)  \(=\) \(13543327580000/27\)

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

All points: \((1 : 0 : 0),\, (1 : -1 : 1),\, (-2 : -1 : 1),\, (-5 : -15 : 3)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 1),\, (-2 : -1 : 1),\, (-5 : -15 : 3)\)
All points: \((1 : 0 : 0),\, (1 : 0 : 1),\, (-2 : 0 : 1),\, (-5 : 0 : 3)\)

magma: [C![-5,-15,3],C![-2,-1,1],C![1,-1,1],C![1,0,0]]; // minimal model
 
magma: [C![-5,0,3],C![-2,0,1],C![1,0,1],C![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(4\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-2 : -1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
\((-2 : -1 : 1) - (1 : 0 : 0)\) \(x + 2z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(3x^2 + xz - 7z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + z^3\) \(0\) \(4\)
Generator $D_0$ Height Order
\((-2 : -1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
\((-2 : -1 : 1) - (1 : 0 : 0)\) \(x + 2z\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(3x^2 + xz - 7z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + z^3\) \(0\) \(4\)
Generator $D_0$ Height Order
\((-2 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(x^2z + xz^2 - 2z^3\) \(0\) \(2\)
\((-2 : 0 : 1) - (1 : 0 : 0)\) \(x + 2z\) \(=\) \(0,\) \(y\) \(=\) \(x^2z + xz^2 - 2z^3\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(3x^2 + xz - 7z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^2z + 3xz^2 + 2z^3\) \(0\) \(4\)

2-torsion field: \(\Q(\sqrt{105}) \)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(3\)
Regulator: \( 1 \)
Real period: \( 19.47088 \)
Tamagawa product: \( 4 \)
Torsion order:\( 16 \)
Leading coefficient: \( 0.304232 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

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 curves:
  Elliptic curve 15.a6
  Elliptic curve 42.a4

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 \(\Q\).