Properties

Label 1385.a.6925.1
Conductor 1385
Discriminant 6925
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

$y^2 + y = x^5 + 3x^4 + 3x^3 - x$ (homogenize, simplify)
$y^2 + z^3y = x^5z + 3x^4z^2 + 3x^3z^3 - xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 + 12x^4 + 12x^3 - 4x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, 3, 3, 1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, 3, 3, 1], R![1]);
 
sage: X = HyperellipticCurve(R([1, -4, 0, 12, 12, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(1385\) = \( 5 \cdot 277 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(6925\) = \( 5^{2} \cdot 277 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(224\) =  \( 2^{5} \cdot 7 \)
\( I_4 \)  = \(-41216\) =  \( - 2^{8} \cdot 7 \cdot 23 \)
\( I_6 \)  = \(-2997760\) =  \( - 2^{9} \cdot 5 \cdot 1171 \)
\( I_{10} \)  = \(28364800\) =  \( 2^{12} \cdot 5^{2} \cdot 277 \)
\( J_2 \)  = \(28\) =  \( 2^{2} \cdot 7 \)
\( J_4 \)  = \(462\) =  \( 2 \cdot 3 \cdot 7 \cdot 11 \)
\( J_6 \)  = \(1916\) =  \( 2^{2} \cdot 479 \)
\( J_8 \)  = \(-39949\) =  \( - 7 \cdot 13 \cdot 439 \)
\( J_{10} \)  = \(6925\) =  \( 5^{2} \cdot 277 \)
\( g_1 \)  = \(17210368/6925\)
\( g_2 \)  = \(10141824/6925\)
\( g_3 \)  = \(1502144/6925\)

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

All points
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((1 : 2 : 1)\)
\((1 : -3 : 1)\) \((4 : 44 : 1)\) \((4 : -45 : 1)\)

magma: [C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,0,0],C![1,2,1],C![4,-45,1],C![4,44,1]];
 

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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) + (1 : 2 : 1) - 2 \cdot(1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + z^3\) \(0.006507\) \(\infty\)

2-torsion field: 5.1.4432.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.006507 \)
Real period: \( 16.82950 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.219037 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(5\) \(2\) \(1\) \(2\) \(( 1 + T )( 1 + T + 5 T^{2} )\)
\(277\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 24 T + 277 T^{2} )\)

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 \(\Q\).