Properties

Label 70450.c.704500.1
Conductor 70450
Discriminant -704500
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

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Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -3, 4, 1, -2], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -3, 4, 1, -2]), R([1, 0, 0, 1]))
 

$y^2 + (x^3 + 1)y = -2x^4 + x^3 + 4x^2 - 3x$

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 70450 \)  =  \( 2 \cdot 5^{2} \cdot 1409 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-704500\)  =  \( -1 \cdot 2^{2} \cdot 5^{3} \cdot 1409 \)

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  =  \(2024\)  =  \( 2^{3} \cdot 11 \cdot 23 \)
\( I_4 \)  =  \(232420\)  =  \( 2^{2} \cdot 5 \cdot 11621 \)
\( I_6 \)  =  \(107744360\)  =  \( 2^{3} \cdot 5 \cdot 2693609 \)
\( I_{10} \)  =  \(-2885632000\)  =  \( -1 \cdot 2^{14} \cdot 5^{3} \cdot 1409 \)
\( J_2 \)  =  \(253\)  =  \( 11 \cdot 23 \)
\( J_4 \)  =  \(246\)  =  \( 2 \cdot 3 \cdot 41 \)
\( J_6 \)  =  \(20576\)  =  \( 2^{5} \cdot 643 \)
\( J_8 \)  =  \(1286303\)  =  \( 1286303 \)
\( J_{10} \)  =  \(-704500\)  =  \( -1 \cdot 2^{2} \cdot 5^{3} \cdot 1409 \)
\( g_1 \)  =  \(-1036579476493/704500\)
\( g_2 \)  =  \(-1991896071/352250\)
\( g_3 \)  =  \(-329262296/176125\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, G2

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) \(C_2 \) (GAP id : [2,1])
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) \(C_2 \) (GAP id : [2,1])

Rational points

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);
 

This curve is locally solvable everywhere.

magma: [C![-3,0,2],C![-3,8,1],C![-3,18,1],C![-3,19,2],C![-1,-16,2],C![-1,-2,1],C![-1,2,1],C![-1,9,2],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-26,3],C![2,-9,3],C![2,-7,1],C![2,-2,1],C![10,-882,3],C![10,-145,3],C![29,-23200,5],C![29,-1314,5]];
 

Known rational points: (-3 : 0 : 2), (-3 : 8 : 1), (-3 : 18 : 1), (-3 : 19 : 2), (-1 : -16 : 2), (-1 : -2 : 1), (-1 : 2 : 1), (-1 : 9 : 2), (0 : -1 : 1), (0 : 0 : 1), (1 : -2 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 0 : 1), (2 : -26 : 3), (2 : -9 : 3), (2 : -7 : 1), (2 : -2 : 1), (10 : -882 : 3), (10 : -145 : 3), (29 : -23200 : 5), (29 : -1314 : 5)

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

Number of rational Weierstrass points: \(0\)

Invariants of the Jacobian:

Analytic rank*: \(3\)

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1);
 

2-Selmer rank: \(4\)

magma: HasSquareSha(Jacobian(C));
 

Order of Ш*: square

Regulator: 0.046457640618

Real period: 16.521293553936620171892200688

Tamagawa numbers: 2 (p = 2), 2 (p = 5), 1 (p = 1409)

magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);
 

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

2-torsion field: splitting field of \(x^{6} - 17 x^{4} - 64 x^{3} - 174 x^{2} - 236 x - 56\) with Galois group $S_4\times C_2$

Sato-Tate group

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

Decomposition

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

Endomorphisms

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 \(\Q\).