Properties

Label 2.0.67.1-323.1-a3
Base field \(\Q(\sqrt{-67}) \)
Conductor norm \( 323 \)
CM no
Base change no
Q-curve no
Torsion order \( 4 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{-67}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x + 17 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([17, -1, 1]))
 
gp: K = nfinit(Polrev([17, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17, -1, 1]);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-2a+17)\) = \((-a)\cdot(a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 323 \) = \(17\cdot19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-1223a+25415)\) = \((-a)^{3}\cdot(a+1)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 640267073 \) = \(17^{3}\cdot19^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{136407521111}{640267073} a + \frac{689979389248}{640267073} \)
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 $\left(-\frac{83156}{243049} a - \frac{114962}{243049} : \frac{10044443}{119823157} a + \frac{316539599}{119823157} : 1\right)$
Height \(6.1008866088633195282241462633401347519\)
Torsion structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(0 : -a - 1 : 1\right)$
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: \( 6.1008866088633195282241462633401347519 \)
Period: \( 2.5234082876283567961329174036992024348 \)
Tamagawa product: \( 12 \)  =  \(3\cdot2^{2}\)
Torsion order: \(4\)
Leading coefficient: \( 5.6424059877744966403715539568564443259 \)
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\((j)_{-}\)
\((-a)\) \(17\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((a+1)\) \(19\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)

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 323.1-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.