Properties

Label 2.2.217.1-12.1-e2
Base field \(\Q(\sqrt{217}) \)
Conductor norm \( 12 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{217}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 54 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-54, -1, 1]))
 
gp: K = nfinit(Polrev([-54, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-54, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+a{y}={x}^{3}+{x}^{2}+\left(-1275546a-8757206\right){x}+2127641751a+14607239178\)
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([0,1]),K([-8757206,-1275546]),K([14607239178,2127641751])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([1,0]),Polrev([0,1]),Polrev([-8757206,-1275546]),Polrev([14607239178,2127641751])], K);
 
magma: E := EllipticCurve([K![1,0],K![1,0],K![0,1],K![-8757206,-1275546],K![14607239178,2127641751]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-104a-714)\) = \((-a+8)\cdot(-a-7)\cdot(-52a-357)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 12 \) = \(2\cdot2\cdot3\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((16a)\) = \((-a+8)^{5}\cdot(-a-7)^{4}\cdot(-52a-357)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -13824 \) = \(-2^{5}\cdot2^{4}\cdot3^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{5784954706489}{864} a + \frac{39716374742807}{864} \)
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 $\left(168 a + 1153 : -248 a - 1699 : 1\right)$
Height \(0.45779655022431382320448510837361855628\)
Torsion structure: trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.45779655022431382320448510837361855628 \)
Period: \( 11.353087017753715668250569879949718304 \)
Tamagawa product: \( 4 \)  =  \(1\cdot2^{2}\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 2.8225822254588866104999987038405994208 \)
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\((j)_{-}\)
\((-a+8)\) \(2\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)
\((-a-7)\) \(2\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((-52a-357)\) \(3\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 12.1-e consists of curves linked by isogenies of degree 3.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.