Properties

Label 2.2.12.1-3600.1-o3
Base field \(\Q(\sqrt{3}) \)
Conductor norm \( 3600 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{3}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 3 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 0, 1]))
 
gp: K = nfinit(Polrev([-3, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}={x}^{3}-{x}^{2}-5442{x}-13050a\)
sage: E = EllipticCurve([K([1,1]),K([-1,0]),K([0,0]),K([-5442,0]),K([0,-13050])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([-1,0]),Polrev([0,0]),Polrev([-5442,0]),Polrev([0,-13050])], K);
 
magma: E := EllipticCurve([K![1,1],K![-1,0],K![0,0],K![-5442,0],K![0,-13050]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((60)\) = \((a+1)^{4}\cdot(a)^{2}\cdot(5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 3600 \) = \(2^{4}\cdot3^{2}\cdot25\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((10125000000000)\) = \((a+1)^{18}\cdot(a)^{8}\cdot(5)^{12}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 102515625000000000000000000 \) = \(2^{18}\cdot3^{8}\cdot25^{12}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{10316097499609}{5859375000} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z\) (no potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \(1\)
Generator $\left(\frac{186705}{1369} a + \frac{531867}{1369} : -\frac{264219366}{50653} a - \frac{371499048}{50653} : 1\right)$
Height \(8.5628377745035658513719595862932789371\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{87}{2} a : -\frac{87}{4} a - \frac{261}{4} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 8.5628377745035658513719595862932789371 \)
Period: \( 0.37362938046915153862936842088716140188 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 3.6942655016410540591106503801499209853 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\((a+1)\) \(2\) \(2\) \(I_{10}^{*}\) Additive \(1\) \(4\) \(18\) \(6\)
\((a)\) \(3\) \(2\) \(I_{2}^{*}\) Additive \(-1\) \(2\) \(8\) \(2\)
\((5)\) \(25\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2B
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 3600.1-o consists of curves linked by isogenies of degrees dividing 12.

Base change

This elliptic curve is a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.