Properties

Label 2.2.53.1-1764.1-c4
Base field \(\Q(\sqrt{53}) \)
Conductor \((42)\)
Conductor norm \( 1764 \)
CM no
Base change yes: 42.a2,117978.h2
Q-curve yes
Torsion order \( 2 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{53}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 13 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-13, -1, 1]))
 
gp: K = nfinit(Pol(Vecrev([-13, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13, -1, 1]);
 

Weierstrass equation

\(y^2+xy+y=x^{3}+x^{2}-914x-10915\)
sage: E = EllipticCurve([K([1,0]),K([1,0]),K([1,0]),K([-914,0]),K([-10915,0])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([-914,0])),Pol(Vecrev([-10915,0]))], K);
 
magma: E := EllipticCurve([K![1,0],K![1,0],K![1,0],K![-914,0],K![-10915,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((42)\) = \( \left(2\right) \cdot \left(3\right) \cdot \left(-a - 2\right) \cdot \left(-a + 3\right) \)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1764 \) = \( 4 \cdot 7^{2} \cdot 9 \)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((933897762)\) = \( \left(2\right) \cdot \left(3\right)^{4} \cdot \left(-a - 2\right)^{8} \cdot \left(-a + 3\right)^{8} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 872165029868608644 \) = \( 4 \cdot 7^{16} \cdot 9^{4} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{84448510979617}{933897762} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{77}{4} : \frac{73}{8} : 1\right)$
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: \( 0.754920939363819 \)
Tamagawa product: \( 16 \)  =  \(2\cdot2\cdot1\cdot2^{2}\)
Torsion order: \(2\)
Leading coefficient: \(1.65914185548301\)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\( \left(-a - 2\right) \) \(7\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)
\( \left(-a + 3\right) \) \(7\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)
\( \left(2\right) \) \(4\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(3\right) \) \(9\) \(4\) \(I_{4}\) 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, 4 and 8.
Its isogeny class 1764.1-c consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base change of elliptic curves 42.a2, 117978.h2, defined over \(\Q\), so it is also a \(\Q\)-curve.