Properties

Base field \(\Q(\sqrt{-7}) \)
Label 2.0.7.1-9801.2-b3
Conductor \((99)\)
Conductor norm \( 9801 \)
CM no
base-change yes: 4851.b3,99.b3
Q-curve yes
Torsion order \( 4 \)
Rank \( 1 \)

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Base field \(\Q(\sqrt{-7}) \)

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

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

Weierstrass equation

\( y^2 + x y + y = x^{3} - x^{2} - 59 x + 186 \)
sage: E = EllipticCurve(K, [1, -1, 1, -59, 186])
 
gp: E = ellinit([1, -1, 1, -59, 186],K)
 
magma: E := ChangeRing(EllipticCurve([1, -1, 1, -59, 186]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

\(\mathfrak{N} \) = \((99)\) = \( \left(3\right)^{2} \cdot \left(-2 a + 3\right) \cdot \left(2 a + 1\right) \)
sage: E.conductor()
 
magma: Conductor(E);
 
\(N(\mathfrak{N}) \) = \( 9801 \) = \( 9^{2} \cdot 11^{2} \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
\(\mathfrak{D}\) = \((216513)\) = \( \left(3\right)^{9} \cdot \left(-2 a + 3\right) \cdot \left(2 a + 1\right) \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
\(N(\mathfrak{D})\) = \( 46877879169 \) = \( 9^{9} \cdot 11^{2} \)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
\(j\) = \( \frac{30664297}{297} \)
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: $\left(-58 : 333 a - 138 : 1\right)$

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

Height: 3.0392116957127087

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

Regulator: 3.03921169571

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

Torsion subgroup

Structure: \(\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Generator: $\left(8 : -18 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(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\((j)_{-}\)
\( \left(-2 a + 3\right) \) \(11\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\( \left(2 a + 1\right) \) \(11\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)
\( \left(3\right) \) \(9\) \(4\) \(I_{3}^*\) Additive \(1\) \(2\) \(9\) \(3\)

Galois Representations

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

prime Image of Galois Representation
\(2\) 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 9801.2-b consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base-change of elliptic curves 4851.b3, 99.b3, defined over \(\Q\), so it is also a \(\Q\)-curve.