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Properties

Label 18289.a.18289.1
Conductor $18289$
Discriminant $18289$
Mordell-Weil group \(\Z \oplus \Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x + 1)y = x^5 - 3x^4 + 2x^2 + 3x$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = x^5z - 3x^4z^2 + 2x^2z^4 + 3xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 - 12x^4 + 9x^2 + 14x + 1$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 3, 2, 0, -3, 1]), R([1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 3, 2, 0, -3, 1], R![1, 1]);
 
Copy content sage:X = HyperellipticCurve(R([1, 14, 9, 0, -12, 4]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(18289\) \(=\) \( 18289 \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(18289\) \(=\) \( 18289 \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(992\) \(=\)  \( 2^{5} \cdot 31 \)
\( I_4 \)  \(=\) \(-16760\) \(=\)  \( - 2^{3} \cdot 5 \cdot 419 \)
\( I_6 \)  \(=\) \(-3877033\) \(=\)  \( -3877033 \)
\( I_{10} \)  \(=\) \(73156\) \(=\)  \( 2^{2} \cdot 18289 \)
\( J_2 \)  \(=\) \(496\) \(=\)  \( 2^{4} \cdot 31 \)
\( J_4 \)  \(=\) \(13044\) \(=\)  \( 2^{2} \cdot 3 \cdot 1087 \)
\( J_6 \)  \(=\) \(328385\) \(=\)  \( 5 \cdot 65677 \)
\( J_8 \)  \(=\) \(-1816744\) \(=\)  \( - 2^{3} \cdot 227093 \)
\( J_{10} \)  \(=\) \(18289\) \(=\)  \( 18289 \)
\( g_1 \)  \(=\) \(30019840638976/18289\)
\( g_2 \)  \(=\) \(1591680221184/18289\)
\( g_3 \)  \(=\) \(80787964160/18289\)

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

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

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 1 : 1)\) \((2 : -1 : 1)\) \((2 : -2 : 1)\)
\((1 : -3 : 1)\) \((4 : 15 : 1)\) \((4 : -20 : 1)\) \((33 : -5148 : 16)\) \((33 : -7396 : 16)\)
Known points
\((1 : 0 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 1 : 1)\) \((2 : -1 : 1)\) \((2 : -2 : 1)\)
\((1 : -3 : 1)\) \((4 : 15 : 1)\) \((4 : -20 : 1)\) \((33 : -5148 : 16)\) \((33 : -7396 : 16)\)
Known points
\((1 : 0 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((2 : -1 : 1)\) \((2 : 1 : 1)\) \((1 : -4 : 1)\)
\((1 : 4 : 1)\) \((4 : -35 : 1)\) \((4 : 35 : 1)\) \((33 : -2248 : 16)\) \((33 : 2248 : 16)\)

Copy content magma:[C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,0,0],C![1,1,1],C![2,-2,1],C![2,-1,1],C![4,-20,1],C![4,15,1],C![33,-7396,16],C![33,-5148,16]]; // minimal model
 
Copy content magma:[C![0,-1,1],C![0,1,1],C![1,-4,1],C![1,0,0],C![1,4,1],C![2,-1,1],C![2,1,1],C![4,-35,1],C![4,35,1],C![33,-2248,16],C![33,2248,16]]; // simplified model
 

Number of rational Weierstrass points: \(1\)

Copy content magma:#Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable everywhere.

Copy content 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: \(\Z \oplus \Z\)

Copy content magma:MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -1 : 1) + (2 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.286541\) \(\infty\)
\((2 : -2 : 1) - (1 : 0 : 0)\) \(x - 2z\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.215169\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -1 : 1) + (2 : -1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.286541\) \(\infty\)
\((2 : -2 : 1) - (1 : 0 : 0)\) \(x - 2z\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.215169\) \(\infty\)
Generator $D_0$ Height Order
\((0 : -1 : 1) + (2 : 1 : 1) - 2 \cdot(1 : 0 : 0)\) \(x (x - 2z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0.286541\) \(\infty\)
\((2 : -1 : 1) - (1 : 0 : 0)\) \(x - 2z\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - 3z^3\) \(0.215169\) \(\infty\)

2-torsion field: 5.1.73156.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.057928 \)
Real period: \( 12.10119 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.700999 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number L-factor Cluster picture Tame reduction?
\(18289\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 104 T + 18289 T^{2} )\) yes

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.6.1 no

Sato-Tate group

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

Decomposition of the Jacobian

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

Copy content magma:HeuristicDecompositionFactors(C);
 

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 \(\Q\).

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

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

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