Properties

Label 40000.e.200000.1
Conductor $40000$
Discriminant $-200000$
Mordell-Weil group \(\Z \times \Z \times \Z/{2}\Z\)
Sato-Tate group $J(C_4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\C)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\mathsf{CM})\)
\(\End(J) \otimes \Q\) \(\mathsf{CM}\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + x^3y = x^5 - 5x^3 - 10x^2 - 8x - 2$ (homogenize, simplify)
$y^2 + x^3y = x^5z - 5x^3z^3 - 10x^2z^4 - 8xz^5 - 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 20x^3 - 40x^2 - 32x - 8$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, -8, -10, -5, 0, 1]), R([0, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, -8, -10, -5, 0, 1], R![0, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-8, -32, -40, -20, 0, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(40000\) \(=\) \( 2^{6} \cdot 5^{4} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-200000\) \(=\) \( - 2^{6} \cdot 5^{5} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(20\) \(=\)  \( 2^{2} \cdot 5 \)
\( I_4 \)  \(=\) \(-20\) \(=\)  \( - 2^{2} \cdot 5 \)
\( I_6 \)  \(=\) \(-40\) \(=\)  \( - 2^{3} \cdot 5 \)
\( I_{10} \)  \(=\) \(8\) \(=\)  \( 2^{3} \)
\( J_2 \)  \(=\) \(100\) \(=\)  \( 2^{2} \cdot 5^{2} \)
\( J_4 \)  \(=\) \(750\) \(=\)  \( 2 \cdot 3 \cdot 5^{3} \)
\( J_6 \)  \(=\) \(-2500\) \(=\)  \( - 2^{2} \cdot 5^{4} \)
\( J_8 \)  \(=\) \(-203125\) \(=\)  \( - 5^{6} \cdot 13 \)
\( J_{10} \)  \(=\) \(200000\) \(=\)  \( 2^{6} \cdot 5^{5} \)
\( g_1 \)  \(=\) \(50000\)
\( g_2 \)  \(=\) \(3750\)
\( g_3 \)  \(=\) \(-125\)

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);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $\GL(2,3)$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : 0 : 1)\) \((-1 : 1 : 1)\) \((-1 : -2 : 2)\) \((-1 : 3 : 2)\)
\((-3 : 11 : 1)\) \((-3 : 16 : 1)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : 0 : 1)\) \((-1 : 1 : 1)\) \((-1 : -2 : 2)\) \((-1 : 3 : 2)\)
\((-3 : 11 : 1)\) \((-3 : 16 : 1)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\) \((-1 : -5 : 2)\) \((-1 : 5 : 2)\)
\((-3 : -5 : 1)\) \((-3 : 5 : 1)\)

magma: [C![-3,11,1],C![-3,16,1],C![-1,-2,2],C![-1,0,1],C![-1,1,1],C![-1,3,2],C![1,-1,0],C![1,0,0]]; // minimal model
 
magma: [C![-3,-5,1],C![-3,5,1],C![-1,-5,2],C![-1,-1,1],C![-1,1,1],C![-1,5,2],C![1,-1,0],C![1,1,0]]; // 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: \(\Z \times \Z \times \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.355997\) \(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.355997\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 2xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - 2z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.355997\) \(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.355997\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 2xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - 2z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3\) \(0.355997\) \(\infty\)
\((-1 : -1 : 1) - (1 : 1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - 2z^3\) \(0.355997\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 + 2xz + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 2xz^2 - 4z^3\) \(0\) \(2\)

2-torsion field: \(\Q(\zeta_{20})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(3\)
Regulator: \( 0.126734 \)
Real period: \( 11.35026 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.719234 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(6\) \(6\) \(1\) \(1 + 2 T + 2 T^{2}\)
\(5\) \(4\) \(5\) \(2\) \(1\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $J(C_4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{U}(1)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 4.4.8000.1 with defining polynomial:
  \(x^{4} - 10 x^{2} + 20\)

Decomposes up to isogeny as the square of the elliptic curve:
  Elliptic curve 4.4.8000.1-25.1-b2

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 \(K = \Q (a) \simeq \) 8.0.64000000.1 with defining polynomial \(x^{8} - 2 x^{6} + 4 x^{4} - 8 x^{2} + 16\)

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

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

\(\End (J_{\overline{\Q}})\)\(\simeq\)a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q(\sqrt{-2}) \)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\C)\)

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{-2}) \) with generator \(-\frac{1}{4} a^{5}\) with minimal polynomial \(x^{2} + 2\):

\(\End (J_{F})\)\(\simeq\)the maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\zeta_{8})\) (CM)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C \times \C\)
  Sato Tate group: $C_4$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-10}) \) with generator \(\frac{1}{4} a^{7} - \frac{1}{4} a^{5} + a^{3}\) with minimal polynomial \(x^{2} + 10\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{-1}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-1}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: $C_{4,1}$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{5}) \) with generator \(\frac{1}{8} a^{6} - \frac{1}{4} a^{4}\) with minimal polynomial \(x^{2} - x - 1\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{-1}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-1}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: $J(C_2)$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-2}, \sqrt{5})\) with generator \(-\frac{1}{8} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3}\) with minimal polynomial \(x^{4} + 6 x^{2} + 4\):

\(\End (J_{F})\)\(\simeq\)the maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\zeta_{8})\) (CM)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C \times \C\)
  Sato Tate group: $C_2$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 4.4.8000.1 with generator \(\frac{1}{8} a^{7} - \frac{1}{4} a^{5} + \frac{1}{2} a^{3} - 2 a\) with minimal polynomial \(x^{4} - 10 x^{2} + 20\):

\(\End (J_{F})\)\(\simeq\)a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)
  Sato Tate group: $C_{2,1}$
  Not of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) \(\Q(\zeta_{5})\) with generator \(\frac{1}{2} a^{2}\) with minimal polynomial \(x^{4} - x^{3} + x^{2} - x + 1\):

\(\End (J_{F})\)\(\simeq\)a maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)the quaternion algebra over \(\Q\) of discriminant 2
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\H\)
  Sato Tate group: $J(C_1)$
  Not of \(\GL_2\)-type, simple