Properties

Label 2.2.85.1-3.1-a3
Base field \(\Q(\sqrt{85}) \)
Conductor \((3,a)\)
Conductor norm \( 3 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{85}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 21 \); class number \(2\).

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 21)
 
gp: K = nfinit(a^2 - a - 21);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-21, -1, 1]);
 

Weierstrass equation

\(y^2+axy=x^{3}+ax^{2}+\left(177a-1059\right)x+2964a-16948\)
sage: E = EllipticCurve(K, [a, a, 0, 177*a - 1059, 2964*a - 16948])
 
gp: E = ellinit([a, a, 0, 177*a - 1059, 2964*a - 16948],K)
 
magma: E := ChangeRing(EllipticCurve([a, a, 0, 177*a - 1059, 2964*a - 16948]),K);
 

This is not a global minimal model: it is minimal at all primes except \((3,a)\). No global minimal model exists.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((3,a)\) = \( \left(3, a\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 3 \) = \( 3 \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((161 a + 2229)\) = \( \left(3, a\right)^{14} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 4782969 \) = \( 3^{14} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
Minimal discriminant: \((a + 3)\) = \( \left(3, a\right)^{2} \)
Minimal discriminant norm: \( 9 \) = \( 3^{2} \)
j-invariant: \( -\frac{392415680105}{9} a + \frac{2007752749790}{9} \)
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: \(0\)
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: $\left(4 a - 8 : 2 a - 42 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 2.79783903083476 \)
Tamagawa product: \( 2 \)
Torsion order: \(2\)
Leading coefficient: \(1.36560712918925\)
Analytic order of Ш: \( 9 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
Primes of good reduction for the curve but which divide the discriminant of the model above (if any) are included.
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(3, a\right) \) \(3\) \(2\) \(I_{2}\) Non-split multiplicative \(1\) \(1\) \(2\) \(2\)

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
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3 and 6.
Its isogeny class 3.1-a consists of curves linked by isogenies of degrees dividing 6.

Base change

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