Properties

Label 12500.b.50000.1
Conductor $12500$
Discriminant $50000$
Mordell-Weil group \(\Z/{5}\Z\)
Sato-Tate group $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{RM}\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = x^5 - 3x^3 + 3x - 2$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = x^5z - 3x^3z^3 + 3xz^5 - 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 10x^3 + 12x - 7$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, 3, 0, -3, 0, 1]), R([1, 0, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, 3, 0, -3, 0, 1], R![1, 0, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([-7, 12, 0, -10, 0, 4, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(12500\) \(=\) \( 2^{2} \cdot 5^{5} \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(50000\) \(=\) \( 2^{4} \cdot 5^{5} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(420\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
\( I_4 \)  \(=\) \(225\) \(=\)  \( 3^{2} \cdot 5^{2} \)
\( I_6 \)  \(=\) \(26865\) \(=\)  \( 3^{3} \cdot 5 \cdot 199 \)
\( I_{10} \)  \(=\) \(2048\) \(=\)  \( 2^{11} \)
\( J_2 \)  \(=\) \(525\) \(=\)  \( 3 \cdot 5^{2} \cdot 7 \)
\( J_4 \)  \(=\) \(11250\) \(=\)  \( 2 \cdot 3^{2} \cdot 5^{4} \)
\( J_6 \)  \(=\) \(322500\) \(=\)  \( 2^{2} \cdot 3 \cdot 5^{4} \cdot 43 \)
\( J_8 \)  \(=\) \(10687500\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 5^{6} \cdot 19 \)
\( J_{10} \)  \(=\) \(50000\) \(=\)  \( 2^{4} \cdot 5^{5} \)
\( g_1 \)  \(=\) \(12762815625/16\)
\( g_2 \)  \(=\) \(260465625/8\)
\( g_3 \)  \(=\) \(7111125/4\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
Copy content magma:AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$
Copy content magma:AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : -1 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : -1 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (1 : 0 : 1)\)

Copy content magma:[C![1,-1,0],C![1,-1,1],C![1,0,0]]; // minimal model
 
Copy content magma:[C![1,-1,0],C![1,0,1],C![1,1,0]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

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

This curve is locally solvable everywhere.

Copy content 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: \(\Z/{5}\Z\)

Copy content magma:MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(5\)
Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0\) \(5\)
Generator $D_0$ Height Order
\((1 : 0 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - z^3\) \(0\) \(5\)

2-torsion field: 5.1.200000.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 5.925134 \)
Tamagawa product: \( 5 \)
Torsion order:\( 5 \)
Leading coefficient: \( 1.185026 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number* L-factor Cluster picture Tame reduction?
\(2\) \(2\) \(4\) \(5\) \(-1^*\) \(( 1 - T )( 1 + T )\) yes
\(5\) \(5\) \(5\) \(1\) \(-1^*\) \(1\) no

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.36.1 no
\(3\) 3.36.1 no

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

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

Copy content magma:HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{5}) \) with defining polynomial \(x^{2} - x - 1\)

Of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{5}}{2}]\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{5}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \R\)

Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

Copy content magma:HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

Copy content magma:HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);