Properties

Label 2.2.301.1-9.2-c2
Base field \(\Q(\sqrt{301}) \)
Conductor norm \( 9 \)
CM yes (\(-7\))
Base change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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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

\({y}^2+\left(a+1\right){x}{y}+{y}={x}^{3}-{x}^{2}+\left(120854a-1108595\right){x}+83293389a-764189269\)
sage: E = EllipticCurve([K([1,1]),K([-1,0]),K([1,0]),K([-1108595,120854]),K([-764189269,83293389])])
 
gp: E = ellinit([Polrev([1,1]),Polrev([-1,0]),Polrev([1,0]),Polrev([-1108595,120854]),Polrev([-764189269,83293389])], K);
 
magma: E := EllipticCurve([K![1,1],K![-1,0],K![1,0],K![-1108595,120854],K![-764189269,83293389]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-17a+156)\) = \((-a+9)^{2}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 9 \) = \(3^{2}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((65a-597)\) = \((-a+9)^{6}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 729 \) = \(3^{6}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -3375 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[(1+\sqrt{-7})/2]\) (potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

Rank: \(1\)
Generator $\left(-\frac{144849812019319}{1463136998404} a + \frac{329208962829337}{365784249601} : -\frac{150929269419324667987}{442453359885868802} a + \frac{5612573161339152936171}{1769813439543475208} : 1\right)$
Height \(16.509866736109127770732593953924070594\)
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: $\left(-99 a + 900 : -351 a + 3262 : 1\right)$
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: \( 16.509866736109127770732593953924070594 \)
Period: \( 5.7094221239649425985397743315811788682 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \( 5.4331597357279157552004276908662377882 \)
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\((j)_{-}\)
\((-a+9)\) \(3\) \(2\) \(I_0^{*}\) Additive \(-1\) \(2\) \(6\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(43\) 43Ns.3.1

For all other primes \(p\), the image is a Borel subgroup if \(p=7\), the normalizer of a split Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -7 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 7 and 14.
Its isogeny class 9.2-c consists of curves linked by isogenies of degrees dividing 14.

Base change

This elliptic curve is a \(\Q\)-curve.

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