Properties

Label 100352.a.100352.1
Conductor 100352
Discriminant 100352
Mordell-Weil group \(\Z/{2}\Z\)
Sato-Tate group $N(G_{3,3})$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type no

Related objects

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Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

Conductor: \( N \)  =  \(100352\) = \( 2^{11} \cdot 7^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(100352\) = \( 2^{11} \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(896\) =  \( 2^{7} \cdot 7 \)
\( I_4 \)  = \(-20480\) =  \( - 2^{12} \cdot 5 \)
\( I_6 \)  = \(-1654784\) =  \( - 2^{14} \cdot 101 \)
\( I_{10} \)  = \(411041792\) =  \( 2^{23} \cdot 7^{2} \)
\( J_2 \)  = \(112\) =  \( 2^{4} \cdot 7 \)
\( J_4 \)  = \(736\) =  \( 2^{5} \cdot 23 \)
\( J_6 \)  = \(-512\) =  \( - 2^{9} \)
\( J_8 \)  = \(-149760\) =  \( - 2^{8} \cdot 3^{2} \cdot 5 \cdot 13 \)
\( J_{10} \)  = \(100352\) =  \( 2^{11} \cdot 7^{2} \)
\( g_1 \)  = \(175616\)
\( g_2 \)  = \(10304\)
\( g_3 \)  = \(-64\)

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

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

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

magma: [C![0,0,1],C![1,0,0]];
 

Number of rational Weierstrass points: \(2\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 4.0.392.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 8.501977 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 2.125494 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(11\) \(11\) \(1\) \(1\)
\(7\) \(2\) \(2\) \(1\) \(1 + T^{2}\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $N(G_{3,3})$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial:
  \(x^{2} + 1\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 2.0.4.1-6272.1-b1
  Elliptic curve 2.0.4.1-6272.1-a1

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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)

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

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

\(\End (J_{\overline{\Q}})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \R\)