Properties

Label 4.4.12400.1-20.2-b2
Base field 4.4.12400.1
Conductor \((-1/2a^3+1/2a^2+7/2a-7/2)\)
Conductor norm \( 20 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 4.4.12400.1

Generator \(a\), with minimal polynomial \( x^{4} - 12 x^{2} + 31 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([31, 0, -12, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([31, 0, -12, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, 0, -12, 0, 1]);
 

Weierstrass equation

\({y}^2+\left(\frac{1}{2}a^{3}-\frac{5}{2}a+1\right){x}{y}+\left(\frac{1}{2}a^{3}-\frac{5}{2}a+1\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{5}{2}a+\frac{5}{2}\right){x}^{2}+\left(\frac{31}{2}a^{3}+\frac{87}{2}a^{2}-\frac{117}{2}a-\frac{327}{2}\right){x}-22a^{3}-64a^{2}+83a+241\)
sage: E = EllipticCurve([K([1,-5/2,0,1/2]),K([5/2,-5/2,-1/2,1/2]),K([1,-5/2,0,1/2]),K([-327/2,-117/2,87/2,31/2]),K([241,83,-64,-22])])
 
gp: E = ellinit([Pol(Vecrev([1,-5/2,0,1/2])),Pol(Vecrev([5/2,-5/2,-1/2,1/2])),Pol(Vecrev([1,-5/2,0,1/2])),Pol(Vecrev([-327/2,-117/2,87/2,31/2])),Pol(Vecrev([241,83,-64,-22]))], K);
 
magma: E := EllipticCurve([K![1,-5/2,0,1/2],K![5/2,-5/2,-1/2,1/2],K![1,-5/2,0,1/2],K![-327/2,-117/2,87/2,31/2],K![241,83,-64,-22]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-1/2a^3+1/2a^2+7/2a-7/2)\) = \((-a+3)\cdot(-1/2a^3-3/2a^2+3/2a+11/2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 20 \) = \(4\cdot5\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-a^3+a^2+7a-7)\) = \((-a+3)^{3}\cdot(-1/2a^3-3/2a^2+3/2a+11/2)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -320 \) = \(4^{3}\cdot5\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{2135237909}{40} a^{3} - \frac{4142533731}{40} a^{2} - \frac{17585965949}{40} a + \frac{34118238951}{40} \)
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 $\left(\frac{1}{2} a^{3} + \frac{5}{2} a^{2} - \frac{3}{2} a - \frac{19}{2} : -\frac{15}{2} a^{3} - \frac{41}{2} a^{2} + \frac{57}{2} a + \frac{155}{2} : 1\right)$
Height \(0.116055896443306\)
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: \( 0.116055896443306 \)
Period: \( 778.174630626015 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 3.24409398320284 \)
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\((j)_{-}\)
\((-a+3)\) \(4\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\((-1/2a^3-3/2a^2+3/2a+11/2)\) \(5\) \(1\) \(I_{1}\) 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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 20.2-b consists of curves linked by isogenies of degree 3.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.