Properties

Label 4.4.10025.1-20.3-a2
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

\({y}^2+\left(\frac{1}{2}a^{3}+\frac{3}{2}a^{2}-\frac{7}{2}a-7\right){x}{y}+{y}={x}^{3}+\left(a^{2}+a-4\right){x}^{2}+\left(-2a^{3}+19a^{2}+20a-124\right){x}-15a^{3}+47a^{2}+148a-416\)
sage: E = EllipticCurve([K([-7,-7/2,3/2,1/2]),K([-4,1,1,0]),K([1,0,0,0]),K([-124,20,19,-2]),K([-416,148,47,-15])])
 
gp: E = ellinit([Polrev([-7,-7/2,3/2,1/2]),Polrev([-4,1,1,0]),Polrev([1,0,0,0]),Polrev([-124,20,19,-2]),Polrev([-416,148,47,-15])], K);
 
magma: E := EllipticCurve([K![-7,-7/2,3/2,1/2],K![-4,1,1,0],K![1,0,0,0],K![-124,20,19,-2],K![-416,148,47,-15]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a^3+3a^2-14a-14)\) = \((-2a^3-3a^2+13a+14)\cdot(a^3+2a^2-7a-9)\)
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: \((-6a^3-7a^2+39a+38)\) = \((-2a^3-3a^2+13a+14)\cdot(a^3+2a^2-7a-9)^{5}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 12500 \) = \(4\cdot5^{5}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -\frac{1574721}{500} a^{3} - \frac{2118087}{500} a^{2} + \frac{424103}{20} a + \frac{843609}{50} \)
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 $\left(a^{3} - 3 a^{2} - 10 a + 26 : 9 a^{3} - 15 a^{2} - 76 a + 144 : 1\right)$
Height \(0.17052133552340748754413221939795261399\)
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: \( 0.17052133552340748754413221939795261399 \)
Period: \( 417.95037533756537300521805086497926393 \)
Tamagawa product: \( 1 \)  =  \(1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 2.84722144221567 \)
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\((j)_{-}\)
\((-2a^3-3a^2+13a+14)\) \(4\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((a^3+2a^2-7a-9)\) \(5\) \(1\) \(I_{5}\) Non-split multiplicative \(1\) \(1\) \(5\) \(5\)

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 20.3-a 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.