Properties

Label 1200.a.450000.1
Conductor 1200
Discriminant -450000
Mordell-Weil group \(\Z/{2}\Z \times \Z/{8}\Z\)
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

$y^2 + (x^3 + x)y = 4x^4 + 25x^2 + 45$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = 4x^4z^2 + 25x^2z^4 + 45z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 18x^4 + 101x^2 + 180$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![45, 0, 25, 0, 4], R![0, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([45, 0, 25, 0, 4]), R([0, 1, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([180, 0, 101, 0, 18, 0, 1]))
 

Invariants

Conductor: \( N \)  =  \(1200\) = \( 2^{4} \cdot 3 \cdot 5^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-450000\) = \( - 2^{4} \cdot 3^{2} \cdot 5^{5} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-72288\) =  \( - 2^{5} \cdot 3^{2} \cdot 251 \)
\( I_4 \)  = \(622464\) =  \( 2^{7} \cdot 3 \cdot 1621 \)
\( I_6 \)  = \(-14918139648\) =  \( - 2^{8} \cdot 3^{2} \cdot 6474887 \)
\( I_{10} \)  = \(-1843200000\) =  \( - 2^{16} \cdot 3^{2} \cdot 5^{5} \)
\( J_2 \)  = \(-9036\) =  \( - 2^{2} \cdot 3^{2} \cdot 251 \)
\( J_4 \)  = \(3395570\) =  \( 2 \cdot 5 \cdot 339557 \)
\( J_6 \)  = \(-1698206400\) =  \( - 2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11 \cdot 71 \cdot 151 \)
\( J_8 \)  = \(953774351375\) =  \( 5^{3} \cdot 227 \cdot 33613193 \)
\( J_{10} \)  = \(-450000\) =  \( - 2^{4} \cdot 3^{2} \cdot 5^{5} \)
\( g_1 \)  = \(418329622965299904/3125\)
\( g_2 \)  = \(3479436045234936/625\)
\( g_3 \)  = \(38515932506304/125\)

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

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (1 : -1 : 0)\)

magma: [C![1,-1,0],C![1,0,0]];
 

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: \(\Z/{2}\Z \times \Z/{8}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 4z^2\) \(=\) \(0,\) \(2y\) \(=\) \(3xz^2\) \(0\) \(2\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(2x^2 + xz + 12z^2\) \(=\) \(0,\) \(4y\) \(=\) \(11xz^2\) \(0\) \(8\)

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

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 5.655371 \)
Tamagawa product: \( 16 \)
Torsion order:\( 16 \)
Leading coefficient: \( 0.353460 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(4\) \(4\) \(1\) \(1 + T + 2 T^{2}\)
\(3\) \(2\) \(1\) \(2\) \(( 1 + T )( 1 + 3 T^{2} )\)
\(5\) \(5\) \(2\) \(8\) \(( 1 - T )^{2}\)

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 80.a2
  Elliptic curve 15.a5

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 \(\Q\).