Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 6300.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6300.a1 | 6300k3 | \([0, 0, 0, -67800, 6276125]\) | \(189123395584/16078125\) | \(2930238281250000\) | \([2]\) | \(41472\) | \(1.7092\) | |
| 6300.a2 | 6300k1 | \([0, 0, 0, -13800, -622375]\) | \(1594753024/4725\) | \(861131250000\) | \([2]\) | \(13824\) | \(1.1599\) | \(\Gamma_0(N)\)-optimal |
| 6300.a3 | 6300k2 | \([0, 0, 0, -8175, -1134250]\) | \(-20720464/178605\) | \(-520812180000000\) | \([2]\) | \(27648\) | \(1.5065\) | |
| 6300.a4 | 6300k4 | \([0, 0, 0, 72825, 28916750]\) | \(14647977776/132355125\) | \(-385947544500000000\) | \([2]\) | \(82944\) | \(2.0558\) |
Rank
sage: E.rank()
The elliptic curves in class 6300.a have rank \(1\).
Complex multiplication
The elliptic curves in class 6300.a do not have complex multiplication.Modular form 6300.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
