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Properties

Label 4.4.10025.1-20.4-b1
Base field 4.4.10025.1
Conductor norm \( 20 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 4.4.10025.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 11 x^{2} + 10 x + 20 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([20, 10, -11, -1, 1]))
 
gp: K = nfinit(Polrev([20, 10, -11, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![20, 10, -11, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}+2a^{2}-7a-10\right){x}{y}+\left(a+1\right){y}={x}^{3}+\left(\frac{1}{2}a^{3}+\frac{3}{2}a^{2}-\frac{5}{2}a-8\right){x}^{2}+\left(-\frac{45}{2}a^{3}-\frac{425}{2}a^{2}+\frac{773}{2}a+610\right){x}-1084a^{3}+1100a^{2}+3210a+929\)
sage: E = EllipticCurve([K([-10,-7,2,1]),K([-8,-5/2,3/2,1/2]),K([1,1,0,0]),K([610,773/2,-425/2,-45/2]),K([929,3210,1100,-1084])])
 
gp: E = ellinit([Polrev([-10,-7,2,1]),Polrev([-8,-5/2,3/2,1/2]),Polrev([1,1,0,0]),Polrev([610,773/2,-425/2,-45/2]),Polrev([929,3210,1100,-1084])], K);
 
magma: E := EllipticCurve([K![-10,-7,2,1],K![-8,-5/2,3/2,1/2],K![1,1,0,0],K![610,773/2,-425/2,-45/2],K![929,3210,1100,-1084]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1/2a^3+1/2a^2-9/2a-2)\) = \((-3/2a^3-3/2a^2+23/2a+8)\cdot(a^3+2a^2-7a-9)\)
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: \((-2a^3-3a^2+13a+17)\) = \((-3/2a^3-3/2a^2+23/2a+8)^{3}\cdot(a^3+2a^2-7a-9)\)
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: \( \frac{3036563556801}{40} a^{3} - \frac{6195482497483}{40} a^{2} - \frac{2695471720915}{4} a + \frac{2919709821181}{2} \)
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 $\left(82 a^{3} + 138 a^{2} - 544 a - 621 : 1583 a^{3} + 2622 a^{2} - 10435 a - 11905 : 1\right)$
Height \(0.14388997525439193839613124998145644510\)
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: \( 0.14388997525439193839613124998145644510 \)
Period: \( 158.36518712213999640928027447585841991 \)
Tamagawa product: \( 3 \)  =  \(3\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 2.73104786387784 \)
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\((j)_{-}\)
\((-3/2a^3-3/2a^2+23/2a+8)\) \(4\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((a^3+2a^2-7a-9)\) \(5\) \(1\) \(I_{1}\) 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
\(5\) 5B.4.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 20.4-b consists of curves linked by isogenies of degree 5.

Base change

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

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