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Properties

Label 12544.g.175616.1
Conductor $12544$
Discriminant $-175616$
Mordell-Weil group \(\Z \oplus \Z/{2}\Z\)
Sato-Tate group $J(E_4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + x^3y = x^5 + x^4 - 2x^2 - 4x - 2$ (homogenize, simplify)
$y^2 + x^3y = x^5z + x^4z^2 - 2x^2z^4 - 4xz^5 - 2z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 + 4x^4 - 8x^2 - 16x - 8$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, -4, -2, 0, 1, 1]), R([0, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, -4, -2, 0, 1, 1], R![0, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-8, -16, -8, 0, 4, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(12544\) \(=\) \( 2^{8} \cdot 7^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-175616\) \(=\) \( - 2^{9} \cdot 7^{3} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(8\) \(=\)  \( 2^{3} \)
\( I_4 \)  \(=\) \(-203\) \(=\)  \( - 7 \cdot 29 \)
\( I_6 \)  \(=\) \(455\) \(=\)  \( 5 \cdot 7 \cdot 13 \)
\( I_{10} \)  \(=\) \(686\) \(=\)  \( 2 \cdot 7^{3} \)
\( J_2 \)  \(=\) \(16\) \(=\)  \( 2^{4} \)
\( J_4 \)  \(=\) \(552\) \(=\)  \( 2^{3} \cdot 3 \cdot 23 \)
\( J_6 \)  \(=\) \(-5632\) \(=\)  \( - 2^{9} \cdot 11 \)
\( J_8 \)  \(=\) \(-98704\) \(=\)  \( - 2^{4} \cdot 31 \cdot 199 \)
\( J_{10} \)  \(=\) \(175616\) \(=\)  \( 2^{9} \cdot 7^{3} \)
\( g_1 \)  \(=\) \(2048/343\)
\( g_2 \)  \(=\) \(4416/343\)
\( g_3 \)  \(=\) \(-2816/343\)

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

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

Rational points

All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (-1 : 1 : 1),\, (-3 : 10 : 1),\, (-3 : 17 : 1)\)
All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-1 : 0 : 1),\, (-1 : 1 : 1),\, (-3 : 10 : 1),\, (-3 : 17 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-1 : -1 : 1),\, (-1 : 1 : 1),\, (-3 : -7 : 1),\, (-3 : 7 : 1)\)

magma: [C![-3,10,1],C![-3,17,1],C![-1,0,1],C![-1,1,1],C![1,-1,0],C![1,0,0]]; // minimal model
 
magma: [C![-3,-7,1],C![-3,7,1],C![-1,-1,1],C![-1,1,1],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: \(\Z \oplus \Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0.046418\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\((-1 : 1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0.046418\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\((-1 : 1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + 2z^3\) \(0.046418\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 2xz^2\) \(0\) \(2\)

2-torsion field: 4.0.392.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.046418 \)
Real period: \( 11.29042 \)
Tamagawa product: \( 6 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.786125 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(8\) \(9\) \(3\) \(1\)
\(7\) \(2\) \(3\) \(2\) \(1 + 7 T^{2}\)

Galois representations

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

Prime \(\ell\) mod-\(\ell\) image
\(2\) 2.90.1
\(3\) 3.270.1

Sato-Tate group

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

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 4.0.2744.1 with defining polynomial:
  \(x^{4} - x^{3} + 3 x^{2} + 3 x + 2\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -56 b^{3} + 112 b^{2} + 280 b + 112\)
  \(g_6 = -812 b^{3} + 8596 b^{2} + 7700 b + 5124\)
   Conductor norm: 1024
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -280 b^{3} + 560 b^{2} - 1288 b + 112\)
  \(g_6 = -16338 b^{3} + 25704 b^{2} - 59150 b - 22764\)
   Conductor norm: 1024

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 \(K = \Q (a) \simeq \) 8.0.481890304.3 with defining polynomial \(x^{8} - 2 x^{7} - 3 x^{6} + 8 x^{5} + x^{4} - 2 x^{3} + 23 x^{2} - 12 x + 2\)

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 \(\frac{1}{2} a^{7} - \frac{7}{8} a^{6} - \frac{7}{4} a^{5} + \frac{7}{2} a^{4} + \frac{7}{4} a^{3} - \frac{7}{8} a^{2} + \frac{21}{2} a - \frac{11}{4}\) 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_4
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{14}) \) with generator \(-\frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{5}{4} a^{4} - \frac{5}{4} a^{3} - \frac{5}{2} a^{2} + \frac{3}{2} a - 4\) with minimal polynomial \(x^{2} - 14\):

\(\End (J_{F})\)\(\simeq\)\(\Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R\)
  Sato Tate group: J(E_2)
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-7}) \) with generator \(\frac{13}{32} a^{7} - \frac{23}{32} a^{6} - \frac{21}{16} a^{5} + \frac{45}{16} a^{4} + \frac{31}{32} a^{3} - \frac{9}{32} a^{2} + \frac{155}{16} a - \frac{33}{16}\) with minimal polynomial \(x^{2} - x + 2\):

\(\End (J_{F})\)\(\simeq\)\(\Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R\)
  Sato Tate group: J(E_2)
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-2}, \sqrt{-7})\) with generator \(\frac{3}{32} a^{7} - \frac{5}{32} a^{6} - \frac{7}{16} a^{5} + \frac{11}{16} a^{4} + \frac{25}{32} a^{3} - \frac{19}{32} a^{2} + \frac{13}{16} a + \frac{5}{16}\) with minimal polynomial \(x^{4} - 2 x^{3} + 9 x^{2} - 8 x + 2\):

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

Over subfield \(F \simeq \) 4.2.21952.1 with generator \(\frac{3}{4} a^{7} - \frac{5}{4} a^{6} - \frac{11}{4} a^{5} + \frac{21}{4} a^{4} + \frac{5}{2} a^{3} - \frac{1}{2} a^{2} + 17 a - 4\) with minimal polynomial \(x^{4} - 2 x^{3} + 5 x^{2} - 4 x - 10\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{2}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: J(E_1)
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 4.0.2744.1 with generator \(-\frac{3}{16} a^{7} + \frac{1}{4} a^{6} + \frac{7}{8} a^{5} - \frac{5}{4} a^{4} - \frac{21}{16} a^{3} + \frac{1}{4} a^{2} - \frac{29}{8} a + \frac{1}{2}\) with minimal polynomial \(x^{4} - x^{3} + 3 x^{2} + 3 x + 2\):

\(\End (J_{F})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: J(E_1)
  Of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 4.0.2744.1 with generator \(\frac{5}{32} a^{7} - \frac{9}{32} a^{6} - \frac{9}{16} a^{5} + \frac{19}{16} a^{4} + \frac{7}{32} a^{3} + \frac{1}{32} a^{2} + \frac{59}{16} a - \frac{23}{16}\) with minimal polynomial \(x^{4} - x^{3} + 3 x^{2} + 3 x + 2\):

\(\End (J_{F})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: J(E_1)
  Of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 4.2.21952.1 with generator \(-\frac{1}{32} a^{7} - \frac{1}{32} a^{6} + \frac{5}{16} a^{5} - \frac{1}{16} a^{4} - \frac{35}{32} a^{3} + \frac{9}{32} a^{2} + \frac{1}{16} a - \frac{15}{16}\) with minimal polynomial \(x^{4} - 2 x^{3} + 5 x^{2} - 4 x - 10\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\sqrt{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{2}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\R \times \R\)
  Sato Tate group: J(E_1)
  Of \(\GL_2\)-type, simple