Properties

Base field \(\Q(\sqrt{17}) \)
Label 2.2.17.1-144.5-c5
Conductor \((-3 a + 15)\)
Conductor norm \( 144 \)
CM no
base-change no
Q-curve yes
Torsion order \( 8 \)
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

\( y^2 + \left(a + 1\right) x y + \left(a + 1\right) y = x^{3} - x^{2} + \left(11 a - 36\right) x - 41 a + 102 \)
magma: E := ChangeRing(EllipticCurve([a + 1, -1, a + 1, 11*a - 36, -41*a + 102]),K);
sage: E = EllipticCurve(K, [a + 1, -1, a + 1, 11*a - 36, -41*a + 102])
gp (2.8): E = ellinit([a + 1, -1, a + 1, 11*a - 36, -41*a + 102],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((-3 a + 15)\) = \( \left(-a - 1\right)^{4} \cdot \left(3\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 144 \) = \( 2^{4} \cdot 9 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((333 a + 207)\) = \( \left(-a - 1\right)^{12} \cdot \left(3\right)^{2} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 331776 \) = \( 2^{12} \cdot 9^{2} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( -\frac{14326000}{9} a + \frac{36913625}{9} \)
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: \(\Z/2\Z\times\Z/4\Z\)
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]
Generators: $\left(-a + 2 : -2 : 1\right)$,$\left(2 a - 5 : -2 : 1\right)$
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\((j)_{-}\)
\( \left(-a - 1\right) \) \(2\) \(4\) \(I_{4}^*\) Additive \(-1\) \(4\) \(12\) \(0\)
\( \left(3\right) \) \(9\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p \) 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, 4 and 8.
Its isogeny class 144.5-c consists of curves linked by isogenies of degrees dividing 16.

Base change

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