Properties

Base field \(\Q(\sqrt{2}) \)
Label 2.2.8.1-578.1-d2
Conductor \((17 a)\)
Conductor norm \( 578 \)
CM no
base-change no
Q-curve yes
Torsion order \( 2 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{2}) \)

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 2)
gp (2.8): K = nfinit(a^2 - 2);

Weierstrass equation

\( y^2 + x y = x^{3} + \left(355 a - 193\right) x - 3135 a + 2315 \)
magma: E := ChangeRing(EllipticCurve([1, 0, 0, 355*a - 193, -3135*a + 2315]),K);
sage: E = EllipticCurve(K, [1, 0, 0, 355*a - 193, -3135*a + 2315])
gp (2.8): E = ellinit([1, 0, 0, 355*a - 193, -3135*a + 2315],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((17 a)\) = \( \left(a\right) \cdot \left(-3 a - 1\right) \cdot \left(3 a - 1\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 578 \) = \( 2 \cdot 17^{2} \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((-693514167 a + 2585878942)\) = \( \left(a\right) \cdot \left(-3 a - 1\right)^{3} \cdot \left(3 a - 1\right)^{12} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 5724846103019631586 \) = \( 2 \cdot 17^{15} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{10008966027980714375}{1165244474459522} a + \frac{7568104598365729000}{582622237229761} \)
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: \( 1 \)
magma: Rank(E);
sage: E.rank()

Generator: $\left(-\frac{347}{98} a + \frac{1381}{98} : -\frac{27289}{1372} a - \frac{10025}{686} : 1\right)$

Height: 5.623384133658726

magma: Generators(E); // includes torsion
sage: E.gens()

Regulator: 5.62338413366

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: $\left(-4 a + \frac{47}{4} : 2 a - \frac{47}{8} : 1\right)$
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()
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(a\right) \) \(2\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\( \left(-3 a - 1\right) \) \(17\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\( \left(3 a - 1\right) \) \(17\) \(2\) \(I_{12}\) Non-split multiplicative \(1\) \(1\) \(12\) \(12\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 578.1-d consists of curves linked by isogenies of degrees dividing 12.

Base change

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