Properties

Label 2.2.113.1-4.1-d4
Base field \(\Q(\sqrt{113}) \)
Conductor norm \( 4 \)
CM no
Base change no
Q-curve yes
Torsion order \( 6 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{113}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 28 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-28, -1, 1]))
 
gp: K = nfinit(Polrev([-28, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-28, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+{y}={x}^{3}+\left(-27992a-134784\right){x}+5873416a+28280926\)
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-134784,-27992]),K([28280926,5873416])])
 
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([1,0]),Polrev([-134784,-27992]),Polrev([28280926,5873416])], K);
 
magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![-134784,-27992],K![28280926,5873416]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2)\) = \((-a+6)\cdot(a+5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 4 \) = \(2\cdot2\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((16a+48)\) = \((-a+6)^{4}\cdot(a+5)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -4096 \) = \(-2^{4}\cdot2^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{85779161}{256} a + \frac{103178791}{64} \)
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: \(\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(5 a + 24 : -646 a - 3111 : 1\right)$
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: \( 34.564848086903415448855566111482425809 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(6\)
Leading coefficient: \( 0.36128748786509891737145651143408869024 \)
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\((j)_{-}\)
\((-a+6)\) \(2\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((a+5)\) \(2\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)

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 and 6.
Its isogeny class 4.1-d consists of curves linked by isogenies of degrees dividing 6.

Base change

This elliptic curve is a \(\Q\)-curve.

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