Properties

Label 2.0.67.1-323.4-a4
Base field \(\Q(\sqrt{-67}) \)
Conductor \((2a+15)\)
Conductor norm \( 323 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
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(Pol(Vecrev([17, -1, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17, -1, 1]);
 

Weierstrass equation

\({y}^2+{x}{y}+\left(a+1\right){y}={x}^3+\left(a+1\right){x}^2+\left(262a+1093\right){x}-2387a+22709\)
sage: E = EllipticCurve([K([1,0]),K([1,1]),K([1,1]),K([1093,262]),K([22709,-2387])])
 
gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([1,1])),Pol(Vecrev([1,1])),Pol(Vecrev([1093,262])),Pol(Vecrev([22709,-2387]))], K);
 
magma: E := EllipticCurve([K![1,0],K![1,1],K![1,1],K![1093,262],K![22709,-2387]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a+15)\) = \((a-1)\cdot(a-2)\)
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: \((-49a-206)\) = \((a-1)^{3}\cdot(a-2)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 93347 \) = \(17^{3}\cdot19\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{35375796455931953}{93347} a + \frac{148925870921969}{5491} \)
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 $\left(\frac{5886137162624093}{1176759339013476} a - \frac{6619700215174163}{588379669506738} : -\frac{60386621978493764348927}{20183732642663596853388} a + \frac{68197912682610114440605}{13455821761775731235592} : 1\right)$
Height \(24.4035464354533\)
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: $\left(5 a - \frac{45}{4} : -3 a + \frac{41}{8} : 1\right)$
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: \( 24.4035464354533 \)
Period: \( 0.630852071907089 \)
Tamagawa product: \( 3 \)  =  \(3\cdot1\)
Torsion order: \(2\)
Leading coefficient: \( 5.64240598777450 \)
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\((j)_{-}\)
\((a-1)\) \(17\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)
\((a-2)\) \(19\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 323.4-a consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.