Show commands:
SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 2100k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2100.k1 | 2100k1 | \([0, 1, 0, -14133, 641988]\) | \(1248870793216/42525\) | \(10631250000\) | \([2]\) | \(2880\) | \(1.0156\) | \(\Gamma_0(N)\)-optimal |
2100.k2 | 2100k2 | \([0, 1, 0, -13508, 701988]\) | \(-68150496976/14467005\) | \(-57868020000000\) | \([2]\) | \(5760\) | \(1.3622\) |
Rank
sage: E.rank()
The elliptic curves in class 2100k have rank \(1\).
Complex multiplication
The elliptic curves in class 2100k do not have complex multiplication.Modular form 2100.2.a.k
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.