Properties

Label 2.0.3.1-102675.1-a5
Base field \(\Q(\sqrt{-3}) \)
Conductor norm \( 102675 \)
CM no
Base change no
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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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

\({y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(35a+166\right){x}-1081a+831\)
sage: E = EllipticCurve([K([1,1]),K([1,-1]),K([0,0]),K([166,35]),K([831,-1081])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([1,-1]),Polrev([0,0]),Polrev([166,35]),Polrev([831,-1081])], K);
 
magma: E := EllipticCurve([K![1,1],K![1,-1],K![0,0],K![166,35],K![831,-1081]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((365a-235)\) = \((-2a+1)\cdot(5)\cdot(-7a+4)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 102675 \) = \(3\cdot25\cdot37^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-6237000a-6917175)\) = \((-2a+1)^{4}\cdot(5)^{2}\cdot(-7a+4)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 129889899455625 \) = \(3^{4}\cdot25^{2}\cdot37^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{13997521}{225} \)
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 $\left(\frac{33}{4} a - 8 : -20 a + \frac{77}{8} : 1\right)$
Height \(3.0181586764526897096703584409307487719\)
Torsion structure: \(\Z/2\Z\oplus\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(10 a - 6 : -7 a + 8 : 1\right)$ $\left(\frac{33}{4} a - 5 : -\frac{23}{4} a + \frac{53}{8} : 1\right)$
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: \( 3.0181586764526897096703584409307487719 \)
Period: \( 0.73509419494234943301860328466568214674 \)
Tamagawa product: \( 16 \)  =  \(2\cdot2\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 5.1237086412940690662013739803697711034 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((5)\) \(25\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((-7a+4)\) \(37\) \(4\) \(I_0^{*}\) Additive \(1\) \(2\) \(6\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 102675.1-a consists of curves linked by isogenies of degrees dividing 16.

Base change

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

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