Properties

Label 4.4.1125.1-961.8-b1
Base field \(\Q(\zeta_{15})^+\)
Conductor norm \( 961 \)
CM yes (\(-15\))
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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Base field \(\Q(\zeta_{15})^+\)

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

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

Weierstrass equation

\({y}^2+\left(a^{2}+a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+2a+1\right){x}^{2}+\left(10406a^{3}+8606a^{2}-25904a-5696\right){x}-1039137a^{3}-859459a^{2}+2586237a+568739\)
sage: E = EllipticCurve([K([-2,1,1,0]),K([1,2,0,-1]),K([-1,-2,1,1]),K([-5696,-25904,8606,10406]),K([568739,2586237,-859459,-1039137])])
 
gp: E = ellinit([Polrev([-2,1,1,0]),Polrev([1,2,0,-1]),Polrev([-1,-2,1,1]),Polrev([-5696,-25904,8606,10406]),Polrev([568739,2586237,-859459,-1039137])], K);
 
magma: E := EllipticCurve([K![-2,1,1,0],K![1,2,0,-1],K![-1,-2,1,1],K![-5696,-25904,8606,10406],K![568739,2586237,-859459,-1039137]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3+4a+5)\) = \((-a^3+5a-2)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 961 \) = \(31^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-93a^3-141a^2+280a+206)\) = \((-a^3+5a-2)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 887503681 \) = \(31^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 85995 a^{3} - 257985 a - 52515 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[(1+\sqrt{-15})/2]\) (potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

Rank: \(1\)
Generator $\left(67 a^{3} + 55 a^{2} - 167 a - 37 : -1342 a^{3} - 1109 a^{2} + 3342 a + 735 : 1\right)$
Height \(0.055608631280997545529478348175807768090\)
Torsion structure: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(24 a^{3} + 19 a^{2} - 61 a - 14 : -37 a^{3} - 30 a^{2} + 93 a + 20 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.055608631280997545529478348175807768090 \)
Period: \( 388.25438335214836852576591004311249721 \)
Tamagawa product: \( 4 \)
Torsion order: \(2\)
Leading coefficient: \( 2.57479290305553 \)
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\((j)_{-}\)
\((-a^3+5a-2)\) \(31\) \(4\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)

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]

For all other primes \(p\), the image is a Borel subgroup if \(p=3\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -15 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -15 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 5, 6, 10, 15 and 30.
Its isogeny class 961.8-b consists of curves linked by isogenies of degrees dividing 30.

Base change

This elliptic curve is a \(\Q\)-curve.

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