Properties

Label 2.0.8.1-11858.2-c4
Base field \(\Q(\sqrt{-2}) \)
Conductor \((77a)\)
Conductor norm \( 11858 \)
CM no
Base change yes: 154.c1,4928.o1
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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Show commands: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{-2}) \)

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([2, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-5164{x}-141529\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([1,0]),K([-5164,0]),K([-141529,0])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([-5164,0])),Pol(Vecrev([-141529,0]))], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![1,0],K![-5164,0],K![-141529,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((77a)\) = \((a)\cdot(a+3)\cdot(a-3)\cdot(7)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 11858 \) = \(2\cdot11\cdot11\cdot49\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((2324168)\) = \((a)^{6}\cdot(a+3)^{2}\cdot(a-3)^{2}\cdot(7)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 5401756892224 \) = \(2^{6}\cdot11^{2}\cdot11^{2}\cdot49^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{15226621995131793}{2324168} \)
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 $\left(-\frac{371}{9} : \frac{2}{27} a + \frac{181}{9} : 1\right)$
Height \(2.25878503693826\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-43 : -14 a + 21 : 1\right)$
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: \( 2.25878503693826 \)
Period: \( 0.285262304200836 \)
Tamagawa product: \( 96 \)  =  \(( 2 \cdot 3 )\cdot2\cdot2\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 5.46745901588000 \)
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\((j)_{-}\)
\((a)\) \(2\) \(6\) \(I_{6}\) Split multiplicative \(-1\) \(1\) \(6\) \(6\)
\((a+3)\) \(11\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((a-3)\) \(11\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)
\((7)\) \(49\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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 and 4.
Its isogeny class 11858.2-c consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base change of 154.c1, 4928.o1, defined over \(\Q\), so it is also a \(\Q\)-curve.