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Properties

Label 5.5.181057.1-25.1-a2
Base field 5.5.181057.1
Conductor norm \( 25 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 5.5.181057.1

Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 4 x^{3} + 7 x^{2} + 2 x - 3 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 2, 7, -4, -2, 1]))
 
gp: K = nfinit(Polrev([-3, 2, 7, -4, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 2, 7, -4, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-3a-1\right){x}{y}+\left(a^{4}-5a^{2}-2a+4\right){y}={x}^{3}+\left(-a^{4}+2a^{3}+3a^{2}-4a\right){x}^{2}+\left(-9a^{4}-3a^{3}+57a^{2}+32a-43\right){x}-50a^{4}-2a^{3}+282a^{2}+118a-194\)
sage: E = EllipticCurve([K([-1,-3,0,1,0]),K([0,-4,3,2,-1]),K([4,-2,-5,0,1]),K([-43,32,57,-3,-9]),K([-194,118,282,-2,-50])])
 
gp: E = ellinit([Polrev([-1,-3,0,1,0]),Polrev([0,-4,3,2,-1]),Polrev([4,-2,-5,0,1]),Polrev([-43,32,57,-3,-9]),Polrev([-194,118,282,-2,-50])], K);
 
magma: E := EllipticCurve([K![-1,-3,0,1,0],K![0,-4,3,2,-1],K![4,-2,-5,0,1],K![-43,32,57,-3,-9],K![-194,118,282,-2,-50]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-2a^2-3a+4)\) = \((a^3-2a^2-3a+4)\)
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: \((-3a^4+16a^2+6a-14)\) = \((a^3-2a^2-3a+4)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -15625 \) = \(-25^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{28069349767}{125} a^{4} - \frac{37403782797}{125} a^{3} - \frac{137248787392}{125} a^{2} + \frac{104873423319}{125} a + \frac{5046062774}{5} \)
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: \(1\)
Generator $\left(-8 a^{4} + 3 a^{3} + 40 a^{2} + 4 a - 20 : 49 a^{4} - 27 a^{3} - 234 a^{2} + 3 a + 100 : 1\right)$
Height \(0.043049739730957349990652348402959580398\)
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: \( 0.043049739730957349990652348402959580398 \)
Period: \( 7209.7565710737452620999051312526351279 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 3.64714875 \)
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\((j)_{-}\)
\((a^3-2a^2-3a+4)\) \(25\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)

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 25.1-a consists of curves linked by isogenies of degree 3.

Base change

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

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