Properties

Label 2.0.11.1-20736.3-bn6
Base field \(\Q(\sqrt{-11}) \)
Conductor norm \( 20736 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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

Generator \(a\), with minimal polynomial \( x^{2} - x + 3 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, -1, 1]))
 
gp: K = nfinit(Polrev([3, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -1, 1]);
 

Weierstrass equation

\({y}^2={x}^{3}-3459{x}-78302\)
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([-3459,0]),K([-78302,0])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,0]),Polrev([-3459,0]),Polrev([-78302,0])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![-3459,0],K![-78302,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((144)\) = \((-a)^{2}\cdot(a-1)^{2}\cdot(2)^{4}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 20736 \) = \(3^{2}\cdot3^{2}\cdot4^{4}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((13436928)\) = \((-a)^{8}\cdot(a-1)^{8}\cdot(2)^{11}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 180551034077184 \) = \(3^{8}\cdot3^{8}\cdot4^{11}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{3065617154}{9} \)
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 $\left(-\frac{9359}{275} : \frac{5796}{15125} a - \frac{2898}{15125} : 1\right)$
Height \(4.9913735268384806360546663570779379592\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-34 : 0 : 1\right)$
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: \( 4.9913735268384806360546663570779379592 \)
Period: \( 0.30294558482761990671354111118756399980 \)
Tamagawa product: \( 16 \)  =  \(2\cdot2\cdot2^{2}\)
Torsion order: \(2\)
Leading coefficient: \( 7.2947151650236540181208052578943657418 \)
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\((j)_{-}\)
\((-a)\) \(3\) \(2\) \(I_{2}^{*}\) Additive \(-1\) \(2\) \(8\) \(2\)
\((a-1)\) \(3\) \(2\) \(I_{2}^{*}\) Additive \(-1\) \(2\) \(8\) \(2\)
\((2)\) \(4\) \(4\) \(I_{3}^{*}\) Additive \(1\) \(4\) \(11\) \(0\)

Galois Representations

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

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4 and 8.
Its isogeny class 20736.3-bn consists of curves linked by isogenies of degrees dividing 8.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 144.b1
\(\Q\) 17424.bu1