Properties

Label 2.0.3.1-71148.2-b3
Base field \(\Q(\sqrt{-3}) \)
Conductor \((-308a+154)\)
Conductor norm \( 71148 \)
CM no
Base change yes: 1386.g2,462.c2
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / 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(Pol(Vecrev([1, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}+a{x}^{2}+\left(-5764a+5763\right){x}+170324\)
sage: E = EllipticCurve([K([1,1]),K([0,1]),K([1,0]),K([5763,-5764]),K([170324,0])])
 
gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,1])),Pol(Vecrev([1,0])),Pol(Vecrev([5763,-5764])),Pol(Vecrev([170324,0]))], K);
 
magma: E := EllipticCurve([K![1,1],K![0,1],K![1,0],K![5763,-5764],K![170324,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-308a+154)\) = \((-2a+1)\cdot(2)\cdot(-3a+1)\cdot(3a-2)\cdot(11)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 71148 \) = \(3\cdot4\cdot7\cdot7\cdot121\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((216872764416)\) = \((-2a+1)^{12}\cdot(2)^{10}\cdot(-3a+1)^{4}\cdot(3a-2)^{4}\cdot(11)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 47033795945437835821056 \) = \(3^{12}\cdot4^{10}\cdot7^{4}\cdot7^{4}\cdot121^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{21184262604460873}{216872764416} \)
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 $\left(-\frac{131}{3} a : \frac{1}{9} a - \frac{5}{9} : 1\right)$
Height \(1.08814943393762\)
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: $\left(-\frac{191}{4} a : \frac{191}{4} a - \frac{195}{8} : 1\right)$ $\left(-41 a : 41 a - 21 : 1\right)$
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: \( 1.08814943393762 \)
Period: \( 0.137317003813298 \)
Tamagawa product: \( 160 \)  =  \(2\cdot( 2 \cdot 5 )\cdot2\cdot2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 3.45073988168233 \)
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\((j)_{-}\)
\((-2a+1)\) \(3\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)
\((2)\) \(4\) \(10\) \(I_{10}\) Split multiplicative \(-1\) \(1\) \(10\) \(10\)
\((-3a+1)\) \(7\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((3a-2)\) \(7\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((11)\) \(121\) \(2\) \(I_{2}\) 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
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 71148.2-b consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base change of 1386.g2, 462.c2, defined over \(\Q\), so it is also a \(\Q\)-curve.