Properties

Label 4.4.17725.1-49.1-a1
Base field 4.4.17725.1
Conductor norm \( 49 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 1 \)

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Base field 4.4.17725.1

Generator \(a\), with minimal polynomial \( x^{4} - 2 x^{3} - 12 x^{2} + 13 x + 41 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([41, 13, -12, -2, 1]))
 
gp: K = nfinit(Polrev([41, 13, -12, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, 13, -12, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{3}-8a-7\right){x}{y}+\left(a^{2}-a-6\right){y}={x}^{3}-{x}^{2}+\left(-316a^{3}+1392a^{2}+609a-5812\right){x}-5945a^{3}+25549a^{2}+12413a-105480\)
sage: E = EllipticCurve([K([-7,-8,0,1]),K([-1,0,0,0]),K([-6,-1,1,0]),K([-5812,609,1392,-316]),K([-105480,12413,25549,-5945])])
 
gp: E = ellinit([Polrev([-7,-8,0,1]),Polrev([-1,0,0,0]),Polrev([-6,-1,1,0]),Polrev([-5812,609,1392,-316]),Polrev([-105480,12413,25549,-5945])], K);
 
magma: E := EllipticCurve([K![-7,-8,0,1],K![-1,0,0,0],K![-6,-1,1,0],K![-5812,609,1392,-316],K![-105480,12413,25549,-5945]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2-10)\) = \((a^2-10)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 49 \) = \(49\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((8053a^3+73613a^2-165832a-447961)\) = \((a^2-10)^{12}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 191581231380566414401 \) = \(49^{12}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{221757824275299}{13841287201} a^{3} + \frac{544265756337879}{13841287201} a^{2} - \frac{93164073307983}{1977326743} a - \frac{1664008534229859}{13841287201} \)
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 $\left(-\frac{74}{9} a^{3} + 13 a^{2} + \frac{442}{9} a - \frac{19}{3} : -\frac{907}{27} a^{3} + \frac{6491}{27} a^{2} - \frac{1840}{27} a - \frac{32182}{27} : 1\right)$
Height \(1.0779552952515073919864183042768190239\)
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: $\left(\frac{11}{4} a^{3} - \frac{21}{4} a^{2} - \frac{31}{2} a + \frac{35}{4} : \frac{37}{8} a^{3} + \frac{9}{8} a^{2} - \frac{317}{8} a - \frac{387}{8} : 1\right)$
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: \( 1.0779552952515073919864183042768190239 \)
Period: \( 19.605972930637165480756659791336255381 \)
Tamagawa product: \( 12 \)
Torsion order: \(2\)
Leading coefficient: \( 1.90492233135250 \)
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\((j)_{-}\)
\((a^2-10)\) \(49\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 49.1-a consists of curves linked by isogenies of degrees dividing 4.

Base change

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

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