Properties

Label 100010.a.400040.1
Conductor 100010
Discriminant -400040
Mordell-Weil group \(\Z \times \Z/{2}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

Learn more about

Show commands for: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = -x^6 + x^5 + 2x^4 + x^3 - 10x^2 + 2x + 5$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = -x^6 + x^5z + 2x^4z^2 + x^3z^3 - 10x^2z^4 + 2xz^5 + 5z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 + 4x^5 + 8x^4 + 6x^3 - 40x^2 + 8x + 21$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![5, 2, -10, 1, 2, 1, -1], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([5, 2, -10, 1, 2, 1, -1]), R([1, 0, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([21, 8, -40, 6, 8, 4, -3]))
 

Invariants

Conductor: \( N \)  =  \(100010\) = \( 2 \cdot 5 \cdot 73 \cdot 137 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-400040\) = \( - 2^{3} \cdot 5 \cdot 73 \cdot 137 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(21736\) =  \( 2^{3} \cdot 11 \cdot 13 \cdot 19 \)
\( I_4 \)  = \(-69788\) =  \( - 2^{2} \cdot 73 \cdot 239 \)
\( I_6 \)  = \(-552591640\) =  \( - 2^{3} \cdot 5 \cdot 59 \cdot 234149 \)
\( I_{10} \)  = \(-1638563840\) =  \( - 2^{15} \cdot 5 \cdot 73 \cdot 137 \)
\( J_2 \)  = \(2717\) =  \( 11 \cdot 13 \cdot 19 \)
\( J_4 \)  = \(308314\) =  \( 2 \cdot 154157 \)
\( J_6 \)  = \(46839264\) =  \( 2^{5} \cdot 3 \cdot 31 \cdot 15739 \)
\( J_8 \)  = \(8051189423\) =  \( 4219 \cdot 1908317 \)
\( J_{10} \)  = \(-400040\) =  \( - 2^{3} \cdot 5 \cdot 73 \cdot 137 \)
\( g_1 \)  = \(-148063561656653357/400040\)
\( g_2 \)  = \(-3091947885524641/200020\)
\( g_3 \)  = \(-43221451942812/50005\)

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

Rational points

All points: \((1 : 0 : 1),\, (1 : -2 : 1),\, (21 : -5666 : 13),\, (21 : -5792 : 13)\)

magma: [C![1,-2,1],C![1,0,1],C![21,-5792,13],C![21,-5666,13]];
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(2 \cdot(1 : 0 : 1) - D_\infty\) \((x - z)^2\) \(=\) \(0,\) \(y\) \(=\) \(-4xz^2 + 4z^3\) \(0.656191\) \(\infty\)
\(D_0 - D_\infty\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)

2-torsion field: splitting field of \(x^{6} - 3 x^{5} - 50 x^{4} + 105 x^{3} + 1525 x^{2} - 1578 x - 19526\) with Galois group $S_4\times C_2$

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.656191 \)
Real period: \( 5.872743 \)
Tamagawa product: \( 3 \)
Torsion order:\( 2 \)
Leading coefficient: \( 2.890231 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(3\) \(1\) \(3\) \(( 1 - T )( 1 - T + 2 T^{2} )\)
\(5\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 5 T^{2} )\)
\(73\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 10 T + 73 T^{2} )\)
\(137\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 10 T + 137 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).