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SageMath
E = EllipticCurve("l1")
E.isogeny_class()
Elliptic curves in class 21840l
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21840.bg4 | 21840l1 | \([0, 1, 0, 9729, -398196]\) | \(6364491337435136/8034291412875\) | \(-128548662606000\) | \([2]\) | \(92160\) | \(1.3933\) | \(\Gamma_0(N)\)-optimal |
| 21840.bg3 | 21840l2 | \([0, 1, 0, -58716, -3902580]\) | \(87450143958975184/25164018140625\) | \(6441988644000000\) | \([2, 2]\) | \(184320\) | \(1.7399\) | |
| 21840.bg2 | 21840l3 | \([0, 1, 0, -351216, 76944420]\) | \(4678944235881273796/202428825314625\) | \(207287117122176000\) | \([2]\) | \(368640\) | \(2.0864\) | |
| 21840.bg1 | 21840l4 | \([0, 1, 0, -861336, -307935036]\) | \(69014771940559650916/9797607421875\) | \(10032750000000000\) | \([2]\) | \(368640\) | \(2.0864\) |
Rank
sage: E.rank()
The elliptic curves in class 21840l have rank \(0\).
Complex multiplication
The elliptic curves in class 21840l do not have complex multiplication.Modular form 21840.2.a.l
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.
