Properties

Base field \(\Q(\sqrt{17}) \)
Label 2.2.17.1-900.1-d8
Conductor \((30)\)
Conductor norm \( 900 \)
CM no
base-change yes: 30.a1,8670.g1
Q-curve yes
Torsion order \( 2 \)
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

\( y^2 + x y + y = x^{3} - 5334 x - 150368 \)
magma: E := ChangeRing(EllipticCurve([1, 0, 1, -5334, -150368]),K);
sage: E = EllipticCurve(K, [1, 0, 1, -5334, -150368])
gp (2.8): E = ellinit([1, 0, 1, -5334, -150368],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

\(\mathfrak{N} \) = \((30)\) = \( \left(-a + 2\right) \cdot \left(-a - 1\right) \cdot \left(3\right) \cdot \left(5\right) \)
magma: Conductor(E);
sage: E.conductor()
\(N(\mathfrak{N}) \) = \( 900 \) = \( 2^{2} \cdot 9 \cdot 25 \)
magma: Norm(Conductor(E));
sage: E.conductor().norm()
\(\mathfrak{D}\) = \((81000)\) = \( \left(-a + 2\right)^{3} \cdot \left(-a - 1\right)^{3} \cdot \left(3\right)^{4} \cdot \left(5\right)^{3} \)
magma: Discriminant(E);
sage: E.discriminant()
gp (2.8): E.disc
\(N(\mathfrak{D})\) = \( 6561000000 \) = \( 2^{6} \cdot 9^{4} \cdot 25^{3} \)
magma: Norm(Discriminant(E));
sage: E.discriminant().norm()
gp (2.8): norm(E.disc)
\(j\) = \( \frac{16778985534208729}{81000} \)
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: $\left(-\frac{169}{4} : \frac{165}{8} : 1\right)$
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\((j)_{-}\)
\( \left(-a + 2\right) \) \(2\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\( \left(-a - 1\right) \) \(2\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\( \left(3\right) \) \(9\) \(4\) \(I_{4}\) Split multiplicative \(-1\) \(1\) \(4\) \(4\)
\( \left(5\right) \) \(25\) \(3\) \(I_{3}\) Split multiplicative \(-1\) \(1\) \(3\) \(3\)

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 900.1-d consists of curves linked by isogenies of degrees dividing 12.

Base change

This curve is the base-change of elliptic curves 30.a1, 8670.g1, defined over \(\Q\), so it is also a \(\Q\)-curve.