Properties

Base field \(\Q(\sqrt{85}) \)
Label 2.2.85.1-12.2-a2
Conductor \((6,2 a + 4)\)
Conductor norm \( 12 \)
CM no
base-change no
Q-curve no
Torsion order \( 2 \)
Rank not available

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

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-21, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 21)
gp (2.8): K = nfinit(a^2 - a - 21);

Weierstrass equation

\( y^2 + a x y + \left(a + 1\right) y = x^{3} + \left(-57679 a - 237048\right) x + 16063422 a + 66017006 \)
magma: E := ChangeRing(EllipticCurve([a, 0, a + 1, -57679*a - 237048, 16063422*a + 66017006]),K);
sage: E = EllipticCurve(K, [a, 0, a + 1, -57679*a - 237048, 16063422*a + 66017006])
gp (2.8): E = ellinit([a, 0, a + 1, -57679*a - 237048, 16063422*a + 66017006],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

\(\mathfrak{N} \) = \((6,2 a + 4)\) = \( \left(2\right) \cdot \left(3, a + 2\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 12 \) = \( 3 \cdot 4 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\((\Delta)\) = \((1062882)\) = \( \left(2\right) \cdot \left(3, a\right)^{12} \cdot \left(3, a + 2\right)^{12} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\Delta)\) = \( 1129718145924 \) = \( 3^{24} \cdot 4 \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(\mathfrak{D}\) = \((388 a + 842)\) = \( \left(2\right) \cdot \left(3, a\right) \cdot \left(3, a + 2\right) \)
\(N(\mathfrak{D})\) = \( 2125764 \) = \( 3^{12} \cdot 4 \)
\(j\) = \( \frac{2087352478285}{531441} a + \frac{5719028825027}{354294} \)
magma: jInvariant(E);
sage: E.j_invariant()
gp (2.8): E.j
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
magma: HasComplexMultiplication(E);
sage: E.has_cm(), E.cm_discriminant()
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank not available.
magma: Rank(E);
sage: E.rank()
magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: not available

magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

Torsion subgroup

Structure: \(\Z/2\Z\)
magma: TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[2]
magma: Order(TorsionSubgroup(E));
sage: E.torsion_order()
gp (2.8): elltors(E)[1]
Generator: $\left(\frac{183}{4} a + \frac{747}{4} : -\frac{467}{4} a - \frac{3847}{8} : 1\right)$
magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);
sage: E.torsion_subgroup().gens()
gp (2.8): elltors(E)[3]

Local data at primes of bad reduction

magma: LocalInformation(E);
sage: E.local_data()
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\((j)_{-}\)
\( \left(3, a\right) \) \(3\) \(1\) \(I_{0}\) Good \(1\) \(0\) \(0\) \(0\)
\( \left(3, a + 2\right) \) \(3\) \(12\) \(I_{12}\) Split multiplicative \(-1\) \(1\) \(12\) \(12\)
\( \left(2\right) \) \(4\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) 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.
Its isogeny class 12.2-a consists of curves linked by isogenies of degree2.

Base change

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