Properties

Base field \(\Q(\sqrt{17}) \)
Label 2.2.17.1-153.1-a1
Conductor \((-6 a + 3)\)
Conductor norm \( 153 \)
CM no
base-change yes: 51.a1,867.c1
Q-curve yes
Torsion order \( 1 \)
Rank not available

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Base field \(\Q(\sqrt{17}) \)

Generator \(a\), with minimal polynomial \( x^{2} - x - 4 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 4)
gp (2.8): K = nfinit(a^2 - a - 4);

Weierstrass equation

\( y^2 + y = x^{3} + x^{2} - 59 x - 196 \)
magma: E := ChangeRing(EllipticCurve([0, 1, 1, -59, -196]),K);
sage: E = EllipticCurve(K, [0, 1, 1, -59, -196])
gp (2.8): E = ellinit([0, 1, 1, -59, -196],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((-6 a + 3)\) = \( \left(3\right) \cdot \left(-2 a + 1\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 153 \) = \( 9 \cdot 17 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((14739)\) = \( \left(3\right) \cdot \left(-2 a + 1\right)^{6} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 217238121 \) = \( 9 \cdot 17^{6} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{23100424192}{14739} \)
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: Trivial
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]

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\((j)_{-}\)
\( \left(-2 a + 1\right) \) \(17\) \(2\) \(I_{6}\) Non-split multiplicative \(1\) \(1\) \(6\) \(6\)
\( \left(3\right) \) \(9\) \(1\) \(I_{1}\) 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
\(3\) 3B.1.2

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3.
Its isogeny class 153.1-a consists of curves linked by isogenies of degree3.

Base change

This curve is the base-change of elliptic curves 51.a1, 867.c1, defined over \(\Q\), so it is also a \(\Q\)-curve.