Properties

Label 2.0.3.1-99372.6-h5
Base field \(\Q(\sqrt{-3}) \)
Conductor \((-322a+14)\)
Conductor norm \( 99372 \)
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(Pol(Vecrev([1, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+a{y}={x}^{3}+\left(-a-1\right){x}^{2}+\left(-1260a+673\right){x}-9991a+15103\)
sage: E = EllipticCurve([K([0,1]),K([-1,-1]),K([0,1]),K([673,-1260]),K([15103,-9991])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,-1])),Pol(Vecrev([0,1])),Pol(Vecrev([673,-1260])),Pol(Vecrev([15103,-9991]))], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,-1],K![0,1],K![673,-1260],K![15103,-9991]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-322a+14)\) = \((-2a+1)\cdot(2)\cdot(-3a+1)\cdot(3a-2)\cdot(4a-3)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 99372 \) = \(3\cdot4\cdot7\cdot7\cdot13^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-160030080a+96081552)\) = \((-2a+1)^{8}\cdot(2)^{4}\cdot(-3a+1)^{2}\cdot(3a-2)^{2}\cdot(4a-3)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 19465352686450944 \) = \(3^{8}\cdot4^{4}\cdot7^{2}\cdot7^{2}\cdot13^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{65597103937}{63504} \)
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 $\left(-15 a - 9 : -22 a + 16 : 1\right)$
Height \(0.316188102753715\)
Torsion structure: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-16 a - 5 : 10 a - 8 : 1\right)$ $\left(-\frac{61}{4} a - \frac{19}{4} : \frac{19}{2} a - \frac{61}{8} : 1\right)$
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: \( 0.316188102753715 \)
Period: \( 0.380020574234020 \)
Tamagawa product: \( 256 \)  =  \(2^{3}\cdot2\cdot2\cdot2\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 4.43988765592712 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\((2)\) \(4\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((-3a+1)\) \(7\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((3a-2)\) \(7\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((4a-3)\) \(13\) \(4\) \(I_0^{*}\) 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 and 4.
Its isogeny class 99372.6-h consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is a \(\Q\)-curve.