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Properties

Label 4.4.12400.1-20.2-d4
Base field 4.4.12400.1
Conductor norm \( 20 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
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(Polrev([31, 0, -12, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31, 0, -12, 0, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+\left(\frac{1}{2}a^{3}-\frac{7}{2}a+1\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-\frac{5}{2}a-\frac{5}{2}\right){x}^{2}+\left(-\frac{341}{2}a^{3}+302a^{2}+\frac{2795}{2}a-2605\right){x}-\frac{4971}{2}a^{3}+4649a^{2}+\frac{40417}{2}a-38557\)
sage: E = EllipticCurve([K([1,0,0,0]),K([-5/2,-5/2,1/2,1/2]),K([1,-7/2,0,1/2]),K([-2605,2795/2,302,-341/2]),K([-38557,40417/2,4649,-4971/2])])
 
gp: E = ellinit([Polrev([1,0,0,0]),Polrev([-5/2,-5/2,1/2,1/2]),Polrev([1,-7/2,0,1/2]),Polrev([-2605,2795/2,302,-341/2]),Polrev([-38557,40417/2,4649,-4971/2])], K);
 
magma: E := EllipticCurve([K![1,0,0,0],K![-5/2,-5/2,1/2,1/2],K![1,-7/2,0,1/2],K![-2605,2795/2,302,-341/2],K![-38557,40417/2,4649,-4971/2]]);
 

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: \((8a^3+24a^2+88a+520)\) = \((-a+3)^{9}\cdot(-1/2a^3-3/2a^2+3/2a+11/2)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 102400000000 \) = \(4^{9}\cdot5^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{4506837263097866069597}{40000} a^{3} - \frac{8743649944588441599153}{40000} a^{2} + \frac{7423723612473221373661}{8000} a + \frac{72013295315208218404901}{40000} \)
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 $\left(\frac{1452}{361} a^{3} - \frac{3673}{361} a^{2} - \frac{13002}{361} a + \frac{29920}{361} : \frac{428179}{13718} a^{3} - \frac{810389}{13718} a^{2} - \frac{3508313}{13718} a + \frac{6644163}{13718} : 1\right)$
Height \(3.0238901219552641476818914916346910789\)
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: $\left(-\frac{3}{2} a^{3} + \frac{3}{2} a^{2} + \frac{19}{2} a - \frac{55}{4} : \frac{1}{2} a^{3} - \frac{3}{4} a^{2} - 3 a + \frac{51}{8} : 1\right)$
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: \( 3.0238901219552641476818914916346910789 \)
Period: \( 11.420599020407286331551508549226395712 \)
Tamagawa product: \( 8 \)  =  \(1\cdot2^{3}\)
Torsion order: \(2\)
Leading coefficient: \( 2.48104153222950 \)
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\((j)_{-}\)
\((-a+3)\) \(4\) \(1\) \(I_{9}\) Non-split multiplicative \(1\) \(1\) \(9\) \(9\)
\((-1/2a^3-3/2a^2+3/2a+11/2)\) \(5\) \(8\) \(I_{8}\) 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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 20.2-d consists of curves linked by isogenies of degrees dividing 6.

Base change

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

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