Properties

Label 5.5.65657.1-25.1-c4
Base field 5.5.65657.1
Conductor norm \( 25 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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Base field 5.5.65657.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 5, 2, -5, -1, 1]))
 
gp: K = nfinit(Polrev([1, 5, 2, -5, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 2, -5, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{4}-a^{3}-4a^{2}+2a+3\right){x}{y}+\left(a^{2}-a-2\right){y}={x}^{3}+\left(-3a^{4}+4a^{3}+13a^{2}-10a-10\right){x}^{2}+\left(4a^{4}-3a^{3}-20a^{2}+5a+15\right){x}-11a^{4}+49a^{2}+27a-4\)
sage: E = EllipticCurve([K([3,2,-4,-1,1]),K([-10,-10,13,4,-3]),K([-2,-1,1,0,0]),K([15,5,-20,-3,4]),K([-4,27,49,0,-11])])
 
gp: E = ellinit([Polrev([3,2,-4,-1,1]),Polrev([-10,-10,13,4,-3]),Polrev([-2,-1,1,0,0]),Polrev([15,5,-20,-3,4]),Polrev([-4,27,49,0,-11])], K);
 
magma: E := EllipticCurve([K![3,2,-4,-1,1],K![-10,-10,13,4,-3],K![-2,-1,1,0,0],K![15,5,-20,-3,4],K![-4,27,49,0,-11]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^4+a^3+3a^2-2a)\) = \((-a^2+a+2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 25 \) = \(5^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((25a^4-36a^3-83a^2+60a+45)\) = \((-a^2+a+2)^{11}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -48828125 \) = \(-5^{11}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{81829946}{3125} a^{4} - \frac{143808649}{3125} a^{3} - \frac{300612723}{3125} a^{2} + \frac{393132181}{3125} a + \frac{109010922}{3125} \)
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 $\left(a^{4} - 4 a^{2} - a + 2 : 3 a^{3} - 10 a - 4 : 1\right)$
Height \(0.044698779245213010973575587299120826177\)
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: \( 0.044698779245213010973575587299120826177 \)
Period: \( 815.81753145448361893618977873435430347 \)
Tamagawa product: \( 4 \)
Torsion order: \(1\)
Leading coefficient: \( 2.84628363 \)
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\((j)_{-}\)
\((-a^2+a+2)\) \(5\) \(4\) \(I_{5}^{*}\) Additive \(1\) \(2\) \(11\) \(5\)

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
\(5\) 5B.4.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3, 5 and 15.
Its isogeny class 25.1-c consists of curves linked by isogenies of degrees dividing 15.

Base change

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

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