Properties

Label 1519.a.1519.1
Conductor 1519
Discriminant -1519
Mordell-Weil group \(\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = -2x^4 + 2x^2 - x - 1$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -2x^4z^2 + 2x^2z^4 - xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 6x^4 + 2x^3 + 9x^2 - 2x - 3$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, -1, 2, 0, -2], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, -1, 2, 0, -2]), R([1, 1, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-3, -2, 9, 2, -6, 0, 1]))
 

Invariants

Conductor: \( N \)  =  \(1519\) = \( 7^{2} \cdot 31 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-1519\) = \( - 7^{2} \cdot 31 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(1608\) =  \( 2^{3} \cdot 3 \cdot 67 \)
\( I_4 \)  = \(7812\) =  \( 2^{2} \cdot 3^{2} \cdot 7 \cdot 31 \)
\( I_6 \)  = \(4136328\) =  \( 2^{3} \cdot 3^{2} \cdot 7 \cdot 29 \cdot 283 \)
\( I_{10} \)  = \(-6221824\) =  \( - 2^{12} \cdot 7^{2} \cdot 31 \)
\( J_2 \)  = \(201\) =  \( 3 \cdot 67 \)
\( J_4 \)  = \(1602\) =  \( 2 \cdot 3^{2} \cdot 89 \)
\( J_6 \)  = \(16160\) =  \( 2^{5} \cdot 5 \cdot 101 \)
\( J_8 \)  = \(170439\) =  \( 3 \cdot 56813 \)
\( J_{10} \)  = \(-1519\) =  \( - 7^{2} \cdot 31 \)
\( g_1 \)  = \(-328080401001/1519\)
\( g_2 \)  = \(-13009202802/1519\)
\( g_3 \)  = \(-652880160/1519\)

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

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: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (-1 : 1 : 1),\, (1 : -1 : 1),\, (1 : -2 : 1)\)

magma: [C![-1,0,1],C![-1,1,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0]];
 

Number of rational Weierstrass points: \(0\)

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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.011132\) \(\infty\)

2-torsion field: 6.0.71528191.2

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.011132 \)
Real period: \( 23.93724 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.266485 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(7\) \(2\) \(2\) \(1\) \(1 + T + 7 T^{2}\)
\(31\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 11 T + 31 T^{2} )\)

Sato-Tate group

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

Decomposition of the Jacobian

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

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 \(\Q\).