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SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 68400bw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68400.dt4 | 68400bw1 | \([0, 0, 0, 4425, 197750]\) | \(3286064/7695\) | \(-22438620000000\) | \([2]\) | \(147456\) | \(1.2458\) | \(\Gamma_0(N)\)-optimal |
| 68400.dt3 | 68400bw2 | \([0, 0, 0, -36075, 2182250]\) | \(445138564/81225\) | \(947408400000000\) | \([2, 2]\) | \(294912\) | \(1.5924\) | |
| 68400.dt2 | 68400bw3 | \([0, 0, 0, -171075, -25222750]\) | \(23735908082/1954815\) | \(45601924320000000\) | \([2]\) | \(589824\) | \(1.9390\) | |
| 68400.dt1 | 68400bw4 | \([0, 0, 0, -549075, 156595250]\) | \(784767874322/35625\) | \(831060000000000\) | \([2]\) | \(589824\) | \(1.9390\) |
Rank
sage: E.rank()
The elliptic curves in class 68400bw have rank \(1\).
Complex multiplication
The elliptic curves in class 68400bw do not have complex multiplication.Modular form 68400.2.a.bw
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
