Properties

Label 8450.c.84500.1
Conductor $8450$
Discriminant $84500$
Mordell-Weil group \(\Z \oplus \Z \oplus \Z/{6}\Z\)
Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = -5x^4 + 10x^3 - 5x^2$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = -5x^4z^2 + 10x^3z^3 - 5x^2z^4$ (dehomogenize, simplify)
$y^2 = x^6 - 20x^4 + 42x^3 - 20x^2 + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -5, 10, -5]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -5, 10, -5], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 0, -20, 42, -20, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(8450\) \(=\) \( 2 \cdot 5^{2} \cdot 13^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(84500\) \(=\) \( 2^{2} \cdot 5^{3} \cdot 13^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1972\) \(=\)  \( 2^{2} \cdot 17 \cdot 29 \)
\( I_4 \)  \(=\) \(60889\) \(=\)  \( 60889 \)
\( I_6 \)  \(=\) \(35769757\) \(=\)  \( 35769757 \)
\( I_{10} \)  \(=\) \(10816000\) \(=\)  \( 2^{9} \cdot 5^{3} \cdot 13^{2} \)
\( J_2 \)  \(=\) \(493\) \(=\)  \( 17 \cdot 29 \)
\( J_4 \)  \(=\) \(7590\) \(=\)  \( 2 \cdot 3 \cdot 5 \cdot 11 \cdot 23 \)
\( J_6 \)  \(=\) \(128000\) \(=\)  \( 2^{10} \cdot 5^{3} \)
\( J_8 \)  \(=\) \(1373975\) \(=\)  \( 5^{2} \cdot 54959 \)
\( J_{10} \)  \(=\) \(84500\) \(=\)  \( 2^{2} \cdot 5^{3} \cdot 13^{2} \)
\( g_1 \)  \(=\) \(29122898485693/84500\)
\( g_2 \)  \(=\) \(90945776163/8450\)
\( g_3 \)  \(=\) \(62220544/169\)

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

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((1 : -2 : 1)\)
\((1 : -4 : 2)\) \((2 : -4 : 1)\) \((1 : -5 : 2)\) \((2 : -5 : 1)\) \((1 : -10 : 3)\) \((3 : -10 : 1)\)
\((1 : -18 : 3)\) \((3 : -18 : 1)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((1 : -2 : 1)\)
\((1 : -4 : 2)\) \((2 : -4 : 1)\) \((1 : -5 : 2)\) \((2 : -5 : 1)\) \((1 : -10 : 3)\) \((3 : -10 : 1)\)
\((1 : -18 : 3)\) \((3 : -18 : 1)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : -1 : 2)\) \((1 : 1 : 2)\)
\((1 : -2 : 1)\) \((1 : 2 : 1)\) \((2 : -1 : 1)\) \((2 : 1 : 1)\) \((1 : -8 : 3)\) \((1 : 8 : 3)\)
\((3 : -8 : 1)\) \((3 : 8 : 1)\)

magma: [C![0,-1,1],C![0,0,1],C![1,-18,3],C![1,-10,3],C![1,-5,2],C![1,-4,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-5,1],C![2,-4,1],C![3,-18,1],C![3,-10,1]]; // minimal model
 
magma: [C![0,-1,1],C![0,1,1],C![1,-8,3],C![1,8,3],C![1,-1,2],C![1,1,2],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,2,1],C![2,-1,1],C![2,1,1],C![3,-8,1],C![3,8,1]]; // 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: \(\Z \oplus \Z \oplus \Z/{6}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((2 : -5 : 1) - (1 : -1 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-5z^3\) \(0.585232\) \(\infty\)
\((1 : -2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.187757\) \(\infty\)
\((0 : 0 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(6\)
Generator $D_0$ Height Order
\((2 : -5 : 1) - (1 : -1 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-5z^3\) \(0.585232\) \(\infty\)
\((1 : -2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.187757\) \(\infty\)
\((0 : 0 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(6\)
Generator $D_0$ Height Order
\((2 : -1 : 1) - (1 : -1 : 0)\) \(z (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 9z^3\) \(0.585232\) \(\infty\)
\((1 : -2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 3z^3\) \(0.187757\) \(\infty\)
\((0 : 1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + z^3\) \(0\) \(6\)

2-torsion field: \(\Q(\sqrt{5}, \sqrt{13})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(3\)
Regulator: \( 0.109881 \)
Real period: \( 20.36995 \)
Tamagawa product: \( 6 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.373046 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + T + 2 T^{2} )\)
\(5\) \(2\) \(3\) \(3\) \(( 1 - T )( 1 + T )\)
\(13\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

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

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.180.4 yes
\(3\) 3.720.4 yes

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 65.a
  Elliptic curve isogeny class 130.a

magma: HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

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

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

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);