Properties

Label 12544.d.25088.1
Conductor $12544$
Discriminant $-25088$
Mordell-Weil group \(\Z \oplus \Z \oplus \Z/{2}\Z\)
Sato-Tate group $E_4$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\mathsf{CM}\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

Downloads

Learn more

Show commands: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

Conductor: \( N \)  \(=\)  \(12544\) \(=\) \( 2^{8} \cdot 7^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-25088\) \(=\) \( - 2^{9} \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(74\) \(=\)  \( 2 \cdot 37 \)
\( I_4 \)  \(=\) \(142\) \(=\)  \( 2 \cdot 71 \)
\( I_6 \)  \(=\) \(3272\) \(=\)  \( 2^{3} \cdot 409 \)
\( I_{10} \)  \(=\) \(98\) \(=\)  \( 2 \cdot 7^{2} \)
\( J_2 \)  \(=\) \(148\) \(=\)  \( 2^{2} \cdot 37 \)
\( J_4 \)  \(=\) \(534\) \(=\)  \( 2 \cdot 3 \cdot 89 \)
\( J_6 \)  \(=\) \(-196\) \(=\)  \( - 2^{2} \cdot 7^{2} \)
\( J_8 \)  \(=\) \(-78541\) \(=\)  \( -78541 \)
\( J_{10} \)  \(=\) \(25088\) \(=\)  \( 2^{9} \cdot 7^{2} \)
\( g_1 \)  \(=\) \(138687914/49\)
\( g_2 \)  \(=\) \(13524351/196\)
\( g_3 \)  \(=\) \(-1369/8\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_4$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $D_4$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((-1 : -2 : 1)\)
\((-1 : 2 : 1)\) \((1 : -4 : 1)\) \((4 : 5 : 3)\) \((-3 : 80 : 4)\) \((-3 : -105 : 4)\) \((4 : -180 : 3)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((-1 : -2 : 1)\)
\((-1 : 2 : 1)\) \((1 : -4 : 1)\) \((4 : 5 : 3)\) \((-3 : 80 : 4)\) \((-3 : -105 : 4)\) \((4 : -180 : 3)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -4 : 1)\) \((-1 : 4 : 1)\)
\((1 : -4 : 1)\) \((1 : 4 : 1)\) \((-3 : -185 : 4)\) \((-3 : 185 : 4)\) \((4 : -185 : 3)\) \((4 : 185 : 3)\)

magma: [C![-3,-105,4],C![-3,80,4],C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-4,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![4,-180,3],C![4,5,3]]; // minimal model
 
magma: [C![-3,-185,4],C![-3,185,4],C![-1,-4,1],C![-1,4,1],C![0,-1,1],C![0,1,1],C![1,-4,1],C![1,-1,0],C![1,1,0],C![1,4,1],C![4,-185,3],C![4,185,3]]; // simplified model
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0.240990\) \(\infty\)
\((1 : -4 : 1) - (1 : 0 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 3z^3\) \(0.240990\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((0 : 0 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0.240990\) \(\infty\)
\((1 : -4 : 1) - (1 : 0 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 3z^3\) \(0.240990\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
Generator $D_0$ Height Order
\((0 : 1 : 1) - (1 : 1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + x^2z + xz^2 + z^3\) \(0.240990\) \(\infty\)
\((1 : -4 : 1) - (1 : 1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + x^2z + xz^2 - 5z^3\) \(0.240990\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + xz^2 + z^3\) \(0\) \(2\)

2-torsion field: 8.0.3211264.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(3\)
Regulator: \( 0.058076 \)
Real period: \( 15.06127 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.437353 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(8\) \(9\) \(2\) \(1 + 2 T + 2 T^{2}\)
\(7\) \(2\) \(2\) \(1\) \(1 + T^{2}\)

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?
\(2\) 2.45.1 yes
\(3\) 3.540.6 no

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $E_4$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{16})^+\) with defining polynomial:
  \(x^{4} - 4 x^{2} + 2\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -7232 b^{3} - 5568 b^{2} + 24704 b + 18976\)
  \(g_6 = 1236992 b^{3} + 945664 b^{2} - 4222208 b - 3231232\)
   Conductor norm: 2401

magma: HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z [\sqrt{-1}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-1}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{16})^+\) with defining polynomial \(x^{4} - 4 x^{2} + 2\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

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

\(\End (J_{\overline{\Q}})\)\(\simeq\)a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{2}) \) with generator \(a^{2} - 2\) with minimal polynomial \(x^{2} - 2\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{-1}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-1}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: E_2
  Of \(\GL_2\)-type, simple

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

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

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