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Properties

Label 16245.c.925965.1
Conductor $16245$
Discriminant $925965$
Mordell-Weil group \(\Z \oplus \Z \oplus \Z/{2}\Z\)
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

$y^2 + (x^3 + x)y = 2x^3 - 2x^2 + 3x$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = 2x^3z^3 - 2x^2z^4 + 3xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^4 + 8x^3 - 7x^2 + 12x$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 3, -2, 2]), R([0, 1, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 3, -2, 2], R![0, 1, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([0, 12, -7, 8, 2, 0, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(16245\) \(=\) \( 3^{2} \cdot 5 \cdot 19^{2} \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(925965\) \(=\) \( 3^{3} \cdot 5 \cdot 19^{3} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(152\) \(=\)  \( 2^{3} \cdot 19 \)
\( I_4 \)  \(=\) \(11692\) \(=\)  \( 2^{2} \cdot 37 \cdot 79 \)
\( I_6 \)  \(=\) \(680631\) \(=\)  \( 3 \cdot 7 \cdot 32411 \)
\( I_{10} \)  \(=\) \(3703860\) \(=\)  \( 2^{2} \cdot 3^{3} \cdot 5 \cdot 19^{3} \)
\( J_2 \)  \(=\) \(76\) \(=\)  \( 2^{2} \cdot 19 \)
\( J_4 \)  \(=\) \(-1708\) \(=\)  \( - 2^{2} \cdot 7 \cdot 61 \)
\( J_6 \)  \(=\) \(-33471\) \(=\)  \( - 3^{2} \cdot 3719 \)
\( J_8 \)  \(=\) \(-1365265\) \(=\)  \( - 5 \cdot 11 \cdot 103 \cdot 241 \)
\( J_{10} \)  \(=\) \(925965\) \(=\)  \( 3^{3} \cdot 5 \cdot 19^{3} \)
\( g_1 \)  \(=\) \(369664/135\)
\( g_2 \)  \(=\) \(-109312/135\)
\( g_3 \)  \(=\) \(-59504/285\)

  (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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((1 : 1 : 1)\) \((1 : -3 : 1)\) \((-3 : 3 : 1)\)
\((3 : 9 : 2)\) \((-3 : 27 : 1)\) \((3 : -48 : 2)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((1 : 1 : 1)\) \((1 : -3 : 1)\) \((-3 : 3 : 1)\)
\((3 : 9 : 2)\) \((-3 : 27 : 1)\) \((3 : -48 : 2)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((1 : -4 : 1)\) \((1 : 4 : 1)\) \((-3 : -24 : 1)\)
\((-3 : 24 : 1)\) \((3 : -57 : 2)\) \((3 : 57 : 2)\)

Copy content magma:[C![-3,3,1],C![-3,27,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,1,1],C![3,-48,2],C![3,9,2]]; // minimal model
 
Copy content magma:[C![-3,-24,1],C![-3,24,1],C![0,0,1],C![1,-4,1],C![1,-1,0],C![1,1,0],C![1,4,1],C![3,-57,2],C![3,57,2]]; // 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: \(\Z \oplus \Z \oplus \Z/{2}\Z\)

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

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz + 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0.712986\) \(\infty\)
\((0 : 0 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.112723\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz + 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + 2z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz + 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0.712986\) \(\infty\)
\((0 : 0 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.112723\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz + 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 + 2z^3\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 - xz + 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + 3xz^2\) \(0.712986\) \(\infty\)
\((0 : 0 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2\) \(0.112723\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 - xz + 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + 3xz^2 + 4z^3\) \(0\) \(2\)

2-torsion field: 6.0.19494000.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(3\)
Regulator: \( 0.078958 \)
Real period: \( 8.544377 \)
Tamagawa product: \( 4 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.674652 \)
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?
\(3\) \(2\) \(3\) \(2\) \(1\) \(( 1 + T )^{2}\) yes
\(5\) \(1\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 3 T + 5 T^{2} )\) yes
\(19\) \(2\) \(3\) \(2\) \(1\) \(( 1 + T )^{2}\) yes

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?
\(2\) 2.60.1 yes
\(3\) 3.80.4 no

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 57.a
  Elliptic curve isogeny class 285.b

Copy content magma:HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(3\) 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 \(\Q\).

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);