Properties

Label 2.2.53.1-900.1-f2
Base field \(\Q(\sqrt{53}) \)
Conductor norm \( 900 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 6 \)
Rank \( 1 \)

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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(Polrev([-13, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13, -1, 1]);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((30)\) = \((2)\cdot(3)\cdot(5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 900 \) = \(4\cdot9\cdot25\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-2160)\) = \((2)^{4}\cdot(3)^{3}\cdot(5)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 4665600 \) = \(4^{4}\cdot9^{3}\cdot25\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{357911}{2160} \)
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 $\left(\frac{113}{49} : -\frac{414}{343} a - \frac{360}{343} : 1\right)$
Height \(4.6341229517284333147217381249456313817\)
Torsion structure: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(3 : 4 : 1\right)$
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: \( 4.6341229517284333147217381249456313817 \)
Period: \( 11.235557132632098287623660143514552130 \)
Tamagawa product: \( 12 \)  =  \(2^{2}\cdot3\cdot1\)
Torsion order: \(6\)
Leading coefficient: \( 4.7679640349432698237877439211904691976 \)
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\((j)_{-}\)
\((2)\) \(4\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\((3)\) \(9\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((5)\) \(25\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 3B.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 900.1-f consists of curves linked by isogenies of degrees dividing 12.

Base change

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

Base field Curve
\(\Q\) 30.a8
\(\Q\) 84270.bb8