Properties

Base field 3.1.23.1
Label 3.1.23.1-259.1-A2
Conductor \((259,-4 a^{2} + 7 a + 1)\)
Conductor norm \( 259 \)
CM no
base-change no
Q-curve no
Torsion order \( 9 \)
Rank \( 0 \)

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Base field 3.1.23.1

Generator \(a\), with minimal polynomial \( x^{3} - x^{2} + 1 \); class number \(1\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^3 - x^2 + 1)
 
gp (2.8): K = nfinit(a^3 - a^2 + 1);
 

Weierstrass equation

\( y^2 + y = x^{3} + a^{2} x^{2} + 9 a x + 8 a^{2} - 15 \)
magma: E := ChangeRing(EllipticCurve([0, a^2, 1, 9*a, 8*a^2 - 15]),K);
 
sage: E = EllipticCurve(K, [0, a^2, 1, 9*a, 8*a^2 - 15])
 
gp (2.8): E = ellinit([0, a^2, 1, 9*a, 8*a^2 - 15],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((259,-4 a^{2} + 7 a + 1)\) = \( \left(2 a^{2} - a\right) \cdot \left(3 a^{2} - a + 1\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 259 \) = \( 7 \cdot 37 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((2044031255371,a + 1984934301353,a^{2} - a + 218241457884)\) = \( \left(2 a^{2} - a\right)^{9} \cdot \left(3 a^{2} - a + 1\right)^{3} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 2044031255371 \) = \( 7^{9} \cdot 37^{3} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{533423413596160}{2044031255371} a^{2} + \frac{14942973020037120}{2044031255371} a + \frac{5291457412308992}{2044031255371} \)
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: \( 0 \)
magma: Rank(E);
 
sage: E.rank()
 
magma: Generators(E); // includes torsion
 
sage: E.gens()
 

Regulator: 1

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

Torsion subgroup

Structure: \(\Z/9\Z\)
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: $\left(10 a^{2} - 21 a + 17 : 74 a^{2} - 131 a + 99 : 1\right)$
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\((j)_{-}\)
\( \left(2 a^{2} - a\right) \) \(7\) \(9\) \(I_{9}\) Split multiplicative \(-1\) \(1\) \(9\) \(9\)
\( \left(3 a^{2} - a + 1\right) \) \(37\) \(3\) \(I_{3}\) 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
\(3\) 3Cs.1.1

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 3 and 9.
Its isogeny class 259.1-A consists of curves linked by isogenies of degrees dividing 27.

Base change

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