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Properties

Label 4.4.16225.1-76.2-c2
Base field 4.4.16225.1
Conductor norm \( 76 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 2 \)

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Base field 4.4.16225.1

Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 13 x^{2} + 6 x + 36 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([36, 6, -13, -1, 1]))
 
gp: K = nfinit(Polrev([36, 6, -13, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36, 6, -13, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a^{2}-a-7\right){x}{y}+\left(\frac{1}{6}a^{3}+\frac{5}{6}a^{2}-\frac{13}{6}a-6\right){y}={x}^{3}+{x}^{2}+\left(\frac{5}{3}a^{3}-\frac{20}{3}a^{2}-\frac{20}{3}a+25\right){x}-\frac{5}{2}a^{3}+\frac{7}{2}a^{2}+\frac{13}{2}a-9\)
sage: E = EllipticCurve([K([-7,-1,1,0]),K([1,0,0,0]),K([-6,-13/6,5/6,1/6]),K([25,-20/3,-20/3,5/3]),K([-9,13/2,7/2,-5/2])])
 
gp: E = ellinit([Polrev([-7,-1,1,0]),Polrev([1,0,0,0]),Polrev([-6,-13/6,5/6,1/6]),Polrev([25,-20/3,-20/3,5/3]),Polrev([-9,13/2,7/2,-5/2])], K);
 
magma: E := EllipticCurve([K![-7,-1,1,0],K![1,0,0,0],K![-6,-13/6,5/6,1/6],K![25,-20/3,-20/3,5/3],K![-9,13/2,7/2,-5/2]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2/3a^3-2/3a^2-17/3a+2)\) = \((1/3a^3-1/3a^2-10/3a+2)\cdot(1/6a^3+5/6a^2-7/6a-5)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 76 \) = \(4\cdot19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((73a^3-193a^2-1210a+1412)\) = \((1/3a^3-1/3a^2-10/3a+2)^{9}\cdot(1/6a^3+5/6a^2-7/6a-5)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -12332795428864 \) = \(-4^{9}\cdot19^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{758824281392413199}{9032809152} a^{3} - \frac{22934531625924522847}{72262473216} a^{2} - \frac{15207761590086801163}{72262473216} a + \frac{26224234042269258567}{24087491072} \)
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: \(2\)
Generators $\left(\frac{1}{8} a^{3} + \frac{7}{8} a^{2} - \frac{9}{8} a - \frac{19}{4} : \frac{5}{12} a^{3} - \frac{1}{24} a^{2} - \frac{37}{24} a - \frac{5}{8} : 1\right)$ $\left(\frac{14180144}{34916281} a^{3} + \frac{5287754}{34916281} a^{2} - \frac{44769429}{34916281} a - \frac{33001693}{34916281} : -\frac{199351705891}{1237921826574} a^{3} + \frac{5329807453393}{1237921826574} a^{2} + \frac{1373091081061}{1237921826574} a - \frac{2361563712242}{206320304429} : 1\right)$
Heights \(1.3844454696262133906095872799868549961\) \(4.9561873509175377838622073143955100041\)
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: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 0.21117097015984349420795811143364241150 \)
Period: \( 97.931487522077370537510312238189895664 \)
Tamagawa product: \( 2 \)  =  \(1\cdot2\)
Torsion order: \(1\)
Leading coefficient: \( 5.19534260198268 \)
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\((j)_{-}\)
\((1/3a^3-1/3a^2-10/3a+2)\) \(4\) \(1\) \(I_{9}\) Non-split multiplicative \(1\) \(1\) \(9\) \(9\)
\((1/6a^3+5/6a^2-7/6a-5)\) \(19\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 76.2-c consists of curves linked by isogenies of degree 3.

Base change

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

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