Properties

Label 4.4.19821.1-25.1-a1
Base field 4.4.19821.1
Conductor norm \( 25 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field 4.4.19821.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 8 x^{2} + 6 x + 3 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([3, 6, -8, -1, 1]))
 
gp: K = nfinit(Polrev([3, 6, -8, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, 6, -8, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-a-4\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(\frac{2}{3}a^{3}-\frac{1}{3}a^{2}-6a+1\right){x}^{2}+\left(3a^{3}+a^{2}-30a+3\right){x}-\frac{22}{3}a^{3}-\frac{1}{3}a^{2}+53a+26\)
sage: E = EllipticCurve([K([-4,-1,1,0]),K([1,-6,-1/3,2/3]),K([1,1,0,0]),K([3,-30,1,3]),K([26,53,-1/3,-22/3])])
 
gp: E = ellinit([Polrev([-4,-1,1,0]),Polrev([1,-6,-1/3,2/3]),Polrev([1,1,0,0]),Polrev([3,-30,1,3]),Polrev([26,53,-1/3,-22/3])], K);
 
magma: E := EllipticCurve([K![-4,-1,1,0],K![1,-6,-1/3,2/3],K![1,1,0,0],K![3,-30,1,3],K![26,53,-1/3,-22/3]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1/3a^3+1/3a^2-3a)\) = \((1/3a^3+1/3a^2-3a)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 25 \) = \(25\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1/3a^3+1/3a^2-3a)\) = \((1/3a^3+1/3a^2-3a)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 25 \) = \(25\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{3609194}{5} a^{3} + \frac{3497552}{5} a^{2} + 3698979 a + \frac{5853102}{5} \)
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: $\left(-\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + 4 a : -\frac{4}{3} a^{3} + \frac{5}{3} a^{2} + 6 a - 1 : 1\right)$
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: \( 1285.3416885176721609331802308482236190 \)
Tamagawa product: \( 1 \)
Torsion order: \(2\)
Leading coefficient: \( 2.28242135208405 \)
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\((j)_{-}\)
\((1/3a^3+1/3a^2-3a)\) \(25\) \(1\) \(I_{1}\) Non-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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 25.1-a consists of curves linked by isogenies of degree 2.

Base change

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

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