Properties

Label 6845.a.6845.1
Conductor 6845
Discriminant 6845
Mordell-Weil group \(\Z \times \Z\)
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + x^3y = x^5 - 7x^3 - 16x^2 - 15x - 5$ (homogenize, simplify)
$y^2 + x^3y = x^5z - 7x^3z^3 - 16x^2z^4 - 15xz^5 - 5z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 28x^3 - 64x^2 - 60x - 20$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-5, -15, -16, -7, 0, 1]), R([0, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-5, -15, -16, -7, 0, 1], R![0, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-20, -60, -64, -28, 0, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(6845\) \(=\) \( 5 \cdot 37^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(6845\) \(=\) \( 5 \cdot 37^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(24\) \(=\)  \( 2^{3} \cdot 3 \)
\( I_4 \)  \(=\) \(852\) \(=\)  \( 2^{2} \cdot 3 \cdot 71 \)
\( I_6 \)  \(=\) \(-14064\) \(=\)  \( - 2^{4} \cdot 3 \cdot 293 \)
\( I_{10} \)  \(=\) \(-27380\) \(=\)  \( - 2^{2} \cdot 5 \cdot 37^{2} \)
\( J_2 \)  \(=\) \(12\) \(=\)  \( 2^{2} \cdot 3 \)
\( J_4 \)  \(=\) \(-136\) \(=\)  \( - 2^{3} \cdot 17 \)
\( J_6 \)  \(=\) \(2040\) \(=\)  \( 2^{3} \cdot 3 \cdot 5 \cdot 17 \)
\( J_8 \)  \(=\) \(1496\) \(=\)  \( 2^{3} \cdot 11 \cdot 17 \)
\( J_{10} \)  \(=\) \(-6845\) \(=\)  \( - 5 \cdot 37^{2} \)
\( g_1 \)  \(=\) \(-248832/6845\)
\( g_2 \)  \(=\) \(235008/6845\)
\( g_3 \)  \(=\) \(-58752/1369\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : 0 : 1)\) \((-1 : 1 : 1)\) \((-2 : 3 : 1)\) \((-2 : 5 : 1)\)
\((3 : -7 : 1)\) \((-3 : 7 : 4)\) \((3 : -20 : 1)\) \((-3 : 20 : 4)\)

magma: [C![-3,7,4],C![-3,20,4],C![-2,3,1],C![-2,5,1],C![-1,0,1],C![-1,1,1],C![1,-1,0],C![1,0,0],C![3,-20,1],C![3,-7,1]];
 

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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.220557\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 3xz + 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2 - 4z^3\) \(0.102222\) \(\infty\)

2-torsion field: 6.2.438080.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.022546 \)
Real period: \( 14.66307 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.330594 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(5\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 5 T^{2} )\)
\(37\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 37.a1
  Elliptic curve 185.b1

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 \(\Q\).