Properties

Label 10110.a.50550.2
Conductor $10110$
Discriminant $-50550$
Mordell-Weil group \(\Z \times \Z/{2}\Z\)
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

Related objects

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Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = 39x^6 - 45x^5 - 62x^4 + 22x^3 + 69x^2 + 50x - 75$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = 39x^6 - 45x^5z - 62x^4z^2 + 22x^3z^3 + 69x^2z^4 + 50xz^5 - 75z^6$ (dehomogenize, simplify)
$y^2 = 156x^6 - 180x^5 - 247x^4 + 90x^3 + 277x^2 + 200x - 300$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-75, 50, 69, 22, -62, -45, 39]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-75, 50, 69, 22, -62, -45, 39], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([-300, 200, 277, 90, -247, -180, 156]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(10110\) \(=\) \( 2 \cdot 3 \cdot 5 \cdot 337 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-50550\) \(=\) \( - 2 \cdot 3 \cdot 5^{2} \cdot 337 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(5467652\) \(=\)  \( 2^{2} \cdot 23 \cdot 103 \cdot 577 \)
\( I_4 \)  \(=\) \(-306863\) \(=\)  \( - 47 \cdot 6529 \)
\( I_6 \)  \(=\) \(-559265218983\) \(=\)  \( - 3^{3} \cdot 20713526629 \)
\( I_{10} \)  \(=\) \(-6470400\) \(=\)  \( - 2^{8} \cdot 3 \cdot 5^{2} \cdot 337 \)
\( J_2 \)  \(=\) \(1366913\) \(=\)  \( 23 \cdot 103 \cdot 577 \)
\( J_4 \)  \(=\) \(77852144018\) \(=\)  \( 2 \cdot 11 \cdot 3538733819 \)
\( J_6 \)  \(=\) \(5912063482701400\) \(=\)  \( 2^{3} \cdot 5^{2} \cdot 337 \cdot 87716075411 \)
\( J_8 \)  \(=\) \(505080025782601398469\) \(=\)  \( 131 \cdot 653 \cdot 10891 \cdot 542135640313 \)
\( J_{10} \)  \(=\) \(-50550\) \(=\)  \( - 2 \cdot 3 \cdot 5^{2} \cdot 337 \)
\( g_1 \)  \(=\) \(-4772043231067501633914862225793/50550\)
\( g_2 \)  \(=\) \(-99417583641640068080475078473/25275\)
\( g_3 \)  \(=\) \(-655572807749456176591436/3\)

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

Known points: \((15 : -2730 : 13)\)
Known points: \((15 : -2730 : 13)\)
Known points: \((15 : 0 : 13)\)

magma: [C![15,-2730,13]]; // minimal model
 
magma: [C![15,0,13]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(2x^2 + 2xz - 5z^2\) \(=\) \(0,\) \(2y\) \(=\) \(41xz^2 - 50z^3\) \(0.303673\) \(\infty\)
\(D_0 - D_\infty\) \(3x^2 - 4z^2\) \(=\) \(0,\) \(6y\) \(=\) \(-3xz^2 - 4z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(2x^2 + 2xz - 5z^2\) \(=\) \(0,\) \(2y\) \(=\) \(41xz^2 - 50z^3\) \(0.303673\) \(\infty\)
\(D_0 - D_\infty\) \(3x^2 - 4z^2\) \(=\) \(0,\) \(6y\) \(=\) \(-3xz^2 - 4z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(2x^2 + 2xz - 5z^2\) \(=\) \(0,\) \(2y\) \(=\) \(x^2z + 83xz^2 - 100z^3\) \(0.303673\) \(\infty\)
\(D_0 - D_\infty\) \(3x^2 - 4z^2\) \(=\) \(0,\) \(6y\) \(=\) \(x^2z - 5xz^2 - 8z^3\) \(0\) \(2\)

2-torsion field: 6.2.3139955712.5

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.303673 \)
Real period: \( 6.345414 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.963467 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - T + 2 T^{2} )\)
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - T + 3 T^{2} )\)
\(5\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + 2 T + 5 T^{2} )\)
\(337\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 14 T + 337 T^{2} )\)

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 \(\Q\).