Properties

Base field \(\Q(\sqrt{-2}) \)
Label 2.0.8.1-5625.2-c1
Conductor \((75)\)
Conductor norm \( 5625 \)
CM no
base-change yes: 75.c1,4800.be1
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{-2}) \)

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

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

Weierstrass equation

\( y^2 + y = x^{3} - x^{2} - 8 x - 7 \)
sage: E = EllipticCurve(K, [0, -1, 1, -8, -7])
 
gp: E = ellinit([0, -1, 1, -8, -7],K)
 
magma: E := ChangeRing(EllipticCurve([0, -1, 1, -8, -7]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((75)\) = \( \left(-a - 1\right) \cdot \left(a - 1\right) \cdot \left(5\right)^{2} \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 5625 \) = \( 3^{2} \cdot 25^{2} \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((1875)\) = \( \left(-a - 1\right) \cdot \left(a - 1\right) \cdot \left(5\right)^{4} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 3515625 \) = \( 3^{2} \cdot 25^{4} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( -\frac{102400}{3} \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
\( \text{End} (E) \) = \(\Z\)   (no Complex Multiplication )
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
\( \text{ST} (E) \) = $\mathrm{SU}(2)$

Mordell-Weil group

Rank: \( 1 \)

sage: E.rank()
 
magma: Rank(E);
 

Generator: $\left(-\frac{1}{2} : \frac{5}{4} a - \frac{1}{2} : 1\right)$

sage: gens = E.gens(); gens
 
magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;
 

Height: 0.3618351555063314

sage: [P.height() for P in gens]
 
magma: [Height(P):P in gens];
 

Regulator: 0.361835155506

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

Torsion subgroup

Structure: Trivial
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord(\(\mathfrak{N}\)) ord(\(\mathfrak{D}\)) ord\((j)_{-}\)
\( \left(-a - 1\right) \) \(3\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\( \left(a - 1\right) \) \(3\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\( \left(5\right) \) \(25\) \(3\) \(IV\) Additive \(1\) \(2\) \(4\) \(0\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.

prime Image of Galois Representation
\(5\) 5B.1.3

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 5.
Its isogeny class 5625.2-c consists of curves linked by isogenies of degree 5.

Base change

This curve is the base-change of elliptic curves 75.c1, 4800.be1, defined over \(\Q\), so it is also a \(\Q\)-curve.