Properties

Label 2.2.301.1-28.1-a4
Base field \(\Q(\sqrt{301}) \)
Conductor norm \( 28 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 2 \)
Rank not available

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Base field \(\Q(\sqrt{301}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 75 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-75, -1, 1]))
 
gp: K = nfinit(Polrev([-75, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-75, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(a+1\right){x}{y}+\left(a+1\right){y}={x}^{3}+{x}^{2}+\left(1059184904a-9717677903\right){x}-45294177781220a+415559396157157\)
sage: E = EllipticCurve([K([1,1]),K([1,0]),K([1,1]),K([-9717677903,1059184904]),K([415559396157157,-45294177781220])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([1,0]),Polrev([1,1]),Polrev([-9717677903,1059184904]),Polrev([415559396157157,-45294177781220])], K);
 
magma: E := EllipticCurve([K![1,1],K![1,0],K![1,1],K![-9717677903,1059184904],K![415559396157157,-45294177781220]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-46a+422)\) = \((2)\cdot(-23a+211)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 28 \) = \(4\cdot7\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((941192)\) = \((2)^{3}\cdot(-23a+211)^{12}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 885842380864 \) = \(4^{3}\cdot7^{12}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{4956477625}{941192} \)
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: \(0 \le r \le 1\)
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: $\left(-\frac{11363}{4} a + 26054 : -\frac{40747}{4} a + \frac{748005}{8} : 1\right)$
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: \(0 \le r \le 1\)
Regulator: not available
Period: \( 7.0277081059000617705423007125374356141 \)
Tamagawa product: \( 6 \)  =  \(3\cdot2\)
Torsion order: \(2\)
Leading coefficient: \( 11.138311968437357700650249118244875965 \)
Analytic order of Ш: not available

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\((j)_{-}\)
\((2)\) \(4\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((-23a+211)\) \(7\) \(2\) \(I_{12}\) Non-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
\(3\) 3Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 28.1-a consists of curves linked by isogenies of degrees dividing 18.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 98.a3
\(\Q\) 25886.d3