Properties

Label 4.4.1125.1-145.3-d1
Base field \(\Q(\zeta_{15})^+\)
Conductor norm \( 145 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
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(Polrev([1, 4, -4, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, -4, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-2\right){x}{y}+\left(a^{3}-3a\right){y}={x}^{3}+{x}^{2}+\left(1238a^{3}-6845a^{2}-20213a-4081\right){x}-124141a^{3}-718644a^{2}-1025357a-185029\)
sage: E = EllipticCurve([K([-2,0,1,0]),K([1,0,0,0]),K([0,-3,0,1]),K([-4081,-20213,-6845,1238]),K([-185029,-1025357,-718644,-124141])])
 
gp: E = ellinit([Polrev([-2,0,1,0]),Polrev([1,0,0,0]),Polrev([0,-3,0,1]),Polrev([-4081,-20213,-6845,1238]),Polrev([-185029,-1025357,-718644,-124141])], K);
 
magma: E := EllipticCurve([K![-2,0,1,0],K![1,0,0,0],K![0,-3,0,1],K![-4081,-20213,-6845,1238],K![-185029,-1025357,-718644,-124141]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^3-a^2+3a-2)\) = \((-a-1)\cdot(-a^3+a^2+4a-2)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 145 \) = \(5\cdot29\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((89517a^3-52832a^2-484425a+296859)\) = \((-a-1)^{2}\cdot(-a^3+a^2+4a-2)^{14}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 7438955816894986587025 \) = \(5^{2}\cdot29^{14}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{522531854629489861064170555139760}{297558232675799463481} a^{3} - \frac{3159218411003952357936457867154706}{1487791163378997317405} a^{2} - \frac{9789549340850333839951979216256637}{1487791163378997317405} a + \frac{12497282500197445579439027620203929}{1487791163378997317405} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(\frac{471}{4} a^{3} + 78 a^{2} - 341 a - \frac{327}{4} : -\frac{365}{8} a^{3} - \frac{297}{8} a^{2} + \frac{887}{8} a + \frac{129}{8} : 1\right)$
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: \( 0.10444775408203468726576502214792519139 \)
Tamagawa product: \( 28 \)  =  \(2\cdot( 2 \cdot 7 )\)
Torsion order: \(2\)
Leading coefficient: \( 1.06811241908055 \)
Analytic order of Ш: \( 49 \) (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\((j)_{-}\)
\((-a-1)\) \(5\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\((-a^3+a^2+4a-2)\) \(29\) \(14\) \(I_{14}\) Split multiplicative \(-1\) \(1\) \(14\) \(14\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(7\) 7B.1.3

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 7 and 14.
Its isogeny class 145.3-d consists of curves linked by isogenies of degrees dividing 14.

Base change

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

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