Properties

Label 4.4.1125.1-45.1-a8
Base field \(\Q(\zeta_{15})^+\)
Conductor \((a^3-5a+1)\)
Conductor norm \( 45 \)
CM no
Base change yes: 45.a1,225.b1
Q-curve yes
Torsion order \( 4 \)
Rank \( 0 \)

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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(Pol(Vecrev([1, 4, -4, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}+a^{2}-2a-2\right){x}{y}+\left(a^{3}+a^{2}-2a-1\right){y}={x}^{3}+\left(-a^{3}+a^{2}+4a-3\right){x}^{2}+\left(2160a^{3}-10801a^{2}+8640a+2163\right){x}-309839a^{3}+1037335a^{2}-687955a-191220\)
sage: E = EllipticCurve([K([-2,-2,1,1]),K([-3,4,1,-1]),K([-1,-2,1,1]),K([2163,8640,-10801,2160]),K([-191220,-687955,1037335,-309839])])
 
gp: E = ellinit([Pol(Vecrev([-2,-2,1,1])),Pol(Vecrev([-3,4,1,-1])),Pol(Vecrev([-1,-2,1,1])),Pol(Vecrev([2163,8640,-10801,2160])),Pol(Vecrev([-191220,-687955,1037335,-309839]))], K);
 
magma: E := EllipticCurve([K![-2,-2,1,1],K![-3,4,1,-1],K![-1,-2,1,1],K![2163,8640,-10801,2160],K![-191220,-687955,1037335,-309839]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^3-5a+1)\) = \((-a-1)\cdot(-a^3+a^2+3a-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 45 \) = \(5\cdot9\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((405)\) = \((-a-1)^{4}\cdot(-a^3+a^2+3a-2)^{8}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 26904200625 \) = \(5^{4}\cdot9^{8}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{1114544804970241}{405} \)
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: \(0\)
Torsion structure: \(\Z/2\Z\times\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(\frac{105}{4} a^{2} - \frac{109}{4} a + 1 : -\frac{105}{4} a^{3} + \frac{307}{8} a + \frac{113}{8} : 1\right)$ $\left(36 a^{3} - 32 a^{2} - 41 a + 1 : -4 a^{3} - 54 a^{2} + 59 a + 21 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 103.875985999069 \)
Tamagawa product: \( 4 \)  =  \(2\cdot2\)
Torsion order: \(4\)
Leading coefficient: \( 0.774245886412445 \)
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\((j)_{-}\)
\((-a-1)\) \(5\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((-a^3+a^2+3a-2)\) \(9\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 4, 8 and 16.
Its isogeny class 45.1-a consists of curves linked by isogenies of degrees dividing 32.

Base change

This curve is the base change of 45.a1, 225.b1, defined over \(\Q\), so it is also a \(\Q\)-curve.