Properties

Label 24649.a.24649.1
Conductor $24649$
Discriminant $24649$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{2}\Z\)
Sato-Tate group $E_6$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\mathsf{CM}\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = x^5 + 4x^4 + 3x^3 - x^2 - x$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^5z + 4x^4z^2 + 3x^3z^3 - x^2z^4 - xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 + 17x^4 + 14x^3 - 3x^2 - 4x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -1, 3, 4, 1]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -1, 3, 4, 1], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([0, -4, -3, 14, 17, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(24649\) \(=\) \( 157^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(24649\) \(=\) \( 157^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(676\) \(=\)  \( 2^{2} \cdot 13^{2} \)
\( I_4 \)  \(=\) \(17113\) \(=\)  \( 109 \cdot 157 \)
\( I_6 \)  \(=\) \(3248173\) \(=\)  \( 17 \cdot 157 \cdot 1217 \)
\( I_{10} \)  \(=\) \(3155072\) \(=\)  \( 2^{7} \cdot 157^{2} \)
\( J_2 \)  \(=\) \(169\) \(=\)  \( 13^{2} \)
\( J_4 \)  \(=\) \(477\) \(=\)  \( 3^{2} \cdot 53 \)
\( J_6 \)  \(=\) \(-467\) \(=\)  \( -467 \)
\( J_8 \)  \(=\) \(-76613\) \(=\)  \( - 23 \cdot 3331 \)
\( J_{10} \)  \(=\) \(24649\) \(=\)  \( 157^{2} \)
\( g_1 \)  \(=\) \(137858491849/24649\)
\( g_2 \)  \(=\) \(2302387893/24649\)
\( g_3 \)  \(=\) \(-13337987/24649\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_6$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $D_6$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

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

magma: [C![-1,0,1],C![0,0,1],C![1,0,0]];
 

Number of rational Weierstrass points: \(3\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(x + z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)

2-torsion field: 3.3.24649.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 23.31292 \)
Tamagawa product: \( 1 \)
Torsion order:\( 4 \)
Leading coefficient: \( 1.457058 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(157\) \(2\) \(2\) \(1\) \(1 - 14 T + 157 T^{2}\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $E_6$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.6.95388992557.1 with defining polynomial:
  \(x^{6} - x^{5} - 65 x^{4} - 160 x^{3} + 20 x^{2} + 208 x + 64\)

Decomposes up to isogeny as the square of the elliptic curve:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -\frac{7769}{12288} b^{5} + \frac{59357}{12288} b^{4} + \frac{46087}{4096} b^{3} + \frac{9527}{3072} b^{2} - \frac{45709}{3072} b - \frac{643}{192}\)
  \(g_6 = \frac{5431415}{73728} b^{5} - \frac{69952763}{147456} b^{4} - \frac{35573845}{16384} b^{3} - \frac{43417879}{147456} b^{2} + \frac{11206189}{4096} b + \frac{33289495}{36864}\)
   Conductor norm: 1

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.95388992557.1 with defining polynomial \(x^{6} - x^{5} - 65 x^{4} - 160 x^{3} + 20 x^{2} + 208 x + 64\)

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

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

\(\End (J_{\overline{\Q}})\)\(\simeq\)an Eichler order of index \(3\) 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 \(F \simeq \) \(\Q(\sqrt{157}) \) with generator \(-\frac{1}{8} a^{5} + \frac{1}{4} a^{4} + \frac{31}{4} a^{3} + \frac{101}{8} a^{2} - \frac{35}{4} a - 10\) with minimal polynomial \(x^{2} - x - 39\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: $E_3$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.3.24649.1 with generator \(\frac{1}{18} a^{5} - \frac{7}{36} a^{4} - \frac{13}{4} a^{3} - \frac{23}{36} a^{2} + \frac{31}{3} a + \frac{20}{9}\) with minimal polynomial \(x^{3} - x^{2} - 52 x - 64\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: $E_2$
  Of \(\GL_2\)-type, simple