Properties

Label 5.5.38569.1-37.1-c2
Base field 5.5.38569.1
Conductor norm \( 37 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 5.5.38569.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 4, 0, -5, 0, 1]))
 
gp: K = nfinit(Polrev([-1, 4, 0, -5, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 4, 0, -5, 0, 1]);
 

Weierstrass equation

\({y}^2+a{y}={x}^{3}+\left(a^{2}-1\right){x}^{2}+\left(11a^{2}-41\right){x}+8a^{4}-61a^{2}+116\)
sage: E = EllipticCurve([K([0,0,0,0,0]),K([-1,0,1,0,0]),K([0,1,0,0,0]),K([-41,0,11,0,0]),K([116,0,-61,0,8])])
 
gp: E = ellinit([Polrev([0,0,0,0,0]),Polrev([-1,0,1,0,0]),Polrev([0,1,0,0,0]),Polrev([-41,0,11,0,0]),Polrev([116,0,-61,0,8])], K);
 
magma: E := EllipticCurve([K![0,0,0,0,0],K![-1,0,1,0,0],K![0,1,0,0,0],K![-41,0,11,0,0],K![116,0,-61,0,8]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^4+a^3-5a^2-5a+4)\) = \((a^4+a^3-5a^2-5a+4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 37 \) = \(37\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((5a^4+3a^3-23a^2-19a+11)\) = \((a^4+a^3-5a^2-5a+4)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 50653 \) = \(37^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{631396984641001648128}{50653} a^{4} + \frac{499267254836904431616}{50653} a^{3} - \frac{2762197178156919664640}{50653} a^{2} - \frac{2184164061589806465024}{50653} a + \frac{798494490468990795776}{50653} \)
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 $\left(-a^{2} + 4 : 0 : 1\right)$
Height \(0.022539312224011562632721802708629837283\)
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: \( 0.022539312224011562632721802708629837283 \)
Period: \( 3460.4072489237849882783993287873625469 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 1.98572310 \)
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\((j)_{-}\)
\((a^4+a^3-5a^2-5a+4)\) \(37\) \(1\) \(I_{3}\) 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 37.1-c 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.