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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 240a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 240.c5 | 240a1 | \([0, -1, 0, -15, -18]\) | \(24918016/45\) | \(720\) | \([2]\) | \(16\) | \(-0.55972\) | \(\Gamma_0(N)\)-optimal |
| 240.c4 | 240a2 | \([0, -1, 0, -20, 0]\) | \(3631696/2025\) | \(518400\) | \([2, 2]\) | \(32\) | \(-0.21315\) | |
| 240.c2 | 240a3 | \([0, -1, 0, -200, 1152]\) | \(868327204/5625\) | \(5760000\) | \([2, 4]\) | \(64\) | \(0.13343\) | |
| 240.c6 | 240a4 | \([0, -1, 0, 80, -80]\) | \(54607676/32805\) | \(-33592320\) | \([2]\) | \(64\) | \(0.13343\) | |
| 240.c1 | 240a5 | \([0, -1, 0, -3200, 70752]\) | \(1770025017602/75\) | \(153600\) | \([4]\) | \(128\) | \(0.48000\) | |
| 240.c3 | 240a6 | \([0, -1, 0, -80, 2400]\) | \(-27995042/1171875\) | \(-2400000000\) | \([4]\) | \(128\) | \(0.48000\) |
Rank
sage: E.rank()
The elliptic curves in class 240a have rank \(0\).
Complex multiplication
The elliptic curves in class 240a do not have complex multiplication.Modular form 240.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
