Properties

Label 2.2.44.1-275.1-g5
Base field \(\Q(\sqrt{11}) \)
Conductor \((5a)\)
Conductor norm \( 275 \)
CM no
Base change yes: 55.a1,9680.r1
Q-curve yes
Torsion order \( 8 \)
Rank \( 0 \)

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Show commands for: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{11}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 11 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-11, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([-11, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11, 0, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}={x}^{3}-{x}^{2}-59{x}+190\)
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,0]),K([-59,0]),K([190,0])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-59,0])),Pol(Vecrev([190,0]))], K);
 
magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,0],K![-59,0],K![190,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((5a)\) = \((a-4)\cdot(-a-4)\cdot(a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 275 \) = \(5\cdot5\cdot11\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((6875)\) = \((a-4)^{4}\cdot(-a-4)^{4}\cdot(a)^{2}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 47265625 \) = \(5^{4}\cdot5^{4}\cdot11^{2}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{22930509321}{6875} \)
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: \(0\)
Torsion structure: \(\Z/2\Z\times\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(2 a - 2 : -a + 1 : 1\right)$ $\left(\frac{7}{2} : -\frac{5}{4} a - \frac{7}{4} : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 16.9305867681545 \)
Tamagawa product: \( 32 \)  =  \(2^{2}\cdot2^{2}\cdot2\)
Torsion order: \(8\)
Leading coefficient: \( 1.27619099523919 \)
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\((j)_{-}\)
\((a-4)\) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((-a-4)\) \(5\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((a)\) \(11\) \(2\) \(I_{2}\) Non-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 and 4.
Its isogeny class 275.1-g consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base change of elliptic curves 55.a1, 9680.r1, defined over \(\Q\), so it is also a \(\Q\)-curve.