Show commands:
SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 21294.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21294.f1 | 21294z3 | \([1, -1, 0, -121653, -16295499]\) | \(124318741396429/51631104\) | \(82693047370752\) | \([2]\) | \(96000\) | \(1.6324\) | |
| 21294.f2 | 21294z4 | \([1, -1, 0, -102933, -21495915]\) | \(-75306487574989/81352871712\) | \(-130295816921271456\) | \([2]\) | \(192000\) | \(1.9789\) | |
| 21294.f3 | 21294z1 | \([1, -1, 0, -4068, 100764]\) | \(4649101309/6804\) | \(10897374852\) | \([2]\) | \(19200\) | \(0.82763\) | \(\Gamma_0(N)\)-optimal |
| 21294.f4 | 21294z2 | \([1, -1, 0, -2898, 159030]\) | \(-1680914269/5786802\) | \(-9268217311626\) | \([2]\) | \(38400\) | \(1.1742\) |
Rank
sage: E.rank()
The elliptic curves in class 21294.f have rank \(1\).
Complex multiplication
The elliptic curves in class 21294.f do not have complex multiplication.Modular form 21294.2.a.f
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
