Properties

Label 2.2.12.1-1734.1-b5
Base field \(\Q(\sqrt{3}) \)
Conductor norm \( 1734 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 8 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{3}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 3 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 0, 1]))
 
gp: K = nfinit(Polrev([-3, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}={x}^{3}-{x}^{2}-1734{x}+27936\)
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([0,0]),K([-1734,0]),K([27936,0])])
 
gp: E = ellinit([Polrev([0,1]),Polrev([-1,0]),Polrev([0,0]),Polrev([-1734,0]),Polrev([27936,0])], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,0],K![0,0],K![-1734,0],K![27936,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((17a+51)\) = \((a+1)\cdot(a)\cdot(17)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1734 \) = \(2\cdot3\cdot289\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((27060804)\) = \((a+1)^{4}\cdot(a)^{8}\cdot(17)^{4}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 732287113126416 \) = \(2^{4}\cdot3^{8}\cdot289^{4}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{576615941610337}{27060804} \)
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 $\left(\frac{73}{2} : -\frac{73}{4} a + \frac{455}{4} : 1\right)$
Height \(3.8269817694884771133715028814573805997\)
Torsion structure: \(\Z/2\Z\oplus\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(24 : -12 a : 1\right)$ $\left(20 : -10 a - 34 : 1\right)$
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: \( 3.8269817694884771133715028814573805997 \)
Period: \( 3.9480248127222485065674715743879541657 \)
Tamagawa product: \( 16 \)  =  \(2\cdot2\cdot2^{2}\)
Torsion order: \(8\)
Leading coefficient: \( 2.1807990443685893313128782856780766879 \)
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\((j)_{-}\)
\((a+1)\) \(2\) \(2\) \(I_{4}\) Non-split multiplicative \(1\) \(1\) \(4\) \(4\)
\((a)\) \(3\) \(2\) \(I_{8}\) Non-split multiplicative \(1\) \(1\) \(8\) \(8\)
\((17)\) \(289\) \(4\) \(I_{4}\) 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
\(2\) 2Cs

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 1734.1-b consists of curves linked by isogenies of degrees dividing 8.

Base change

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

Base field Curve
\(\Q\) 306.b2
\(\Q\) 816.b2