Properties

Label 2.2.12.1-1800.1-i2
Base field \(\Q(\sqrt{3}) \)
Conductor norm \( 1800 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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Show commands: Magma / PariGP / SageMath

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

\({y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(a-1\right){x}^{2}+\left(-124a-215\right){x}+871a+1511\)
sage: E = EllipticCurve([K([1,1]),K([-1,1]),K([1,1]),K([-215,-124]),K([1511,871])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([-1,1]),Polrev([1,1]),Polrev([-215,-124]),Polrev([1511,871])], K);
 
magma: E := EllipticCurve([K![1,1],K![-1,1],K![1,1],K![-215,-124],K![1511,871]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((30a+30)\) = \((a+1)^{3}\cdot(a)^{2}\cdot(5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1800 \) = \(2^{3}\cdot3^{2}\cdot25\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((21600)\) = \((a+1)^{10}\cdot(a)^{6}\cdot(5)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 466560000 \) = \(2^{10}\cdot3^{6}\cdot25^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{3721734}{25} \)
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 $\left(4 a + 9 : -16 a - 26 : 1\right)$
Height \(0.19091958455252588286144010851337123072\)
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: $\left(\frac{5}{2} a + 6 : -\frac{19}{4} a - \frac{29}{4} : 1\right)$
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: \( 0.19091958455252588286144010851337123072 \)
Period: \( 14.770184929637810605164012769049506861 \)
Tamagawa product: \( 8 \)  =  \(2\cdot2\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 3.2561603368765979851729395908030226240 \)
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\((j)_{-}\)
\((a+1)\) \(2\) \(2\) \(III^{*}\) Additive \(-1\) \(3\) \(10\) \(0\)
\((a)\) \(3\) \(2\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)
\((5)\) \(25\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 1800.1-i consists of curves linked by isogenies of degree 2.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 360.d1
\(\Q\) 720.a1