Properties

Base field \(\Q(\sqrt{10}) \)
Label 2.2.40.1-90.1-f12
Conductor \((-3 a)\)
Conductor norm \( 90 \)
CM no
base-change no
Q-curve yes
Torsion order \( 2 \)
Rank not available

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Base field \(\Q(\sqrt{10}) \)

Generator \(a\), with minimal polynomial \( x^{2} - 10 \); class number \(2\).

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 10)
 
gp (2.8): K = nfinit(a^2 - 10);
 

Weierstrass equation

\( y^2 + x y + y = x^{3} + \left(-12650 a - 45334\right) x - 1494440 a - 4876368 \)
magma: E := ChangeRing(EllipticCurve([1, 0, 1, -12650*a - 45334, -1494440*a - 4876368]),K);
 
sage: E = EllipticCurve(K, [1, 0, 1, -12650*a - 45334, -1494440*a - 4876368])
 
gp (2.8): E = ellinit([1, 0, 1, -12650*a - 45334, -1494440*a - 4876368],K)
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((-3 a)\) = \( \left(2, a\right) \cdot \left(3, a + 1\right) \cdot \left(3, a + 2\right) \cdot \left(5, a\right) \)
magma: Conductor(E);
 
sage: E.conductor()
 
\(N(\mathfrak{N}) \) = \( 90 \) = \( 2 \cdot 3^{2} \cdot 5 \)
magma: Norm(Conductor(E));
 
sage: E.conductor().norm()
 
\(\mathfrak{D}\) = \((-3330 a + 7200)\) = \( \left(2, a\right)^{3} \cdot \left(3, a + 1\right)^{8} \cdot \left(3, a + 2\right)^{2} \cdot \left(5, a\right)^{3} \)
magma: Discriminant(E);
 
sage: E.discriminant()
 
gp (2.8): E.disc
 
\(N(\mathfrak{D})\) = \( 59049000 \) = \( 2^{3} \cdot 3^{10} \cdot 5^{3} \)
magma: Norm(Discriminant(E));
 
sage: E.discriminant().norm()
 
gp (2.8): norm(E.disc)
 
\(j\) = \( \frac{4451448983765985658895227933}{656100} a + \frac{140767176767424108251921384}{6561} \)
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: $\left(-40 a - \frac{169}{4} : 20 a + \frac{165}{8} : 1\right)$
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\((j)_{-}\)
\( \left(2, a\right) \) \(2\) \(1\) \(I_{3}\) Non-split multiplicative \(1\) \(1\) \(3\) \(3\)
\( \left(3, a + 1\right) \) \(3\) \(8\) \(I_{8}\) Split multiplicative \(-1\) \(1\) \(8\) \(8\)
\( \left(3, a + 2\right) \) \(3\) \(2\) \(I_{2}\) Split multiplicative \(-1\) \(1\) \(2\) \(2\)
\( \left(5, a\right) \) \(5\) \(1\) \(I_{3}\) Non-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, 8, 12 and 24.
Its isogeny class 90.1-f consists of curves linked by isogenies of degrees dividing 24.

Base change

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