Properties

Label 2.0.3.1-12348.2-a9
Base field \(\Q(\sqrt{-3}) \)
Conductor norm \( 12348 \)
CM no
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

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

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

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

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}+a{x}^{2}+\left(6408a-8048\right){x}-269145a+201566\)
sage: E = EllipticCurve([K([1,0]),K([0,1]),K([1,0]),K([-8048,6408]),K([201566,-269145])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,1]),Polrev([1,0]),Polrev([-8048,6408]),Polrev([201566,-269145])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,1],K![1,0],K![-8048,6408],K![201566,-269145]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-126a+42)\) = \((-2a+1)^{2}\cdot(2)\cdot(-3a+1)^{2}\cdot(3a-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 12348 \) = \(3^{2}\cdot4\cdot7^{2}\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-718226460a+331874172)\) = \((-2a+1)^{6}\cdot(2)^{2}\cdot(-3a+1)^{15}\cdot(3a-2)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 387628902163826064 \) = \(3^{6}\cdot4^{2}\cdot7^{15}\cdot7\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{10722436976428375}{161414428} a + \frac{3017980745593000}{40353607} \)
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 $\left(38 a + 111 : 876 a + 775 : 1\right)$
Height \(1.3185793410033461829324056070527130982\)
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: $\left(-24 a + 57 : 12 a - 29 : 1\right)$
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: \( 1.3185793410033461829324056070527130982 \)
Period: \( 0.19103168031847237239004799552012849179 \)
Tamagawa product: \( 16 \)  =  \(2\cdot2\cdot2^{2}\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 2.3268640946961499941607468424048285161 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(2\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)
\((2)\) \(4\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((-3a+1)\) \(7\) \(4\) \(I_{9}^{*}\) Additive \(-1\) \(2\) \(15\) \(9\)
\((3a-2)\) \(7\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

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[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 6, 9 and 18.
Its isogeny class 12348.2-a consists of curves linked by isogenies of degrees dividing 18.

Base change

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

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