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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 20160.ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20160.ca1 | 20160bl4 | \([0, 0, 0, -134508, 18987568]\) | \(5633270409316/14175\) | \(677221171200\) | \([2]\) | \(65536\) | \(1.5084\) | |
| 20160.ca2 | 20160bl3 | \([0, 0, 0, -23628, -1023248]\) | \(30534944836/8203125\) | \(391910400000000\) | \([2]\) | \(65536\) | \(1.5084\) | |
| 20160.ca3 | 20160bl2 | \([0, 0, 0, -8508, 289168]\) | \(5702413264/275625\) | \(3292047360000\) | \([2, 2]\) | \(32768\) | \(1.1618\) | |
| 20160.ca4 | 20160bl1 | \([0, 0, 0, 312, 17512]\) | \(4499456/180075\) | \(-134425267200\) | \([2]\) | \(16384\) | \(0.81521\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 20160.ca have rank \(1\).
Complex multiplication
The elliptic curves in class 20160.ca do not have complex multiplication.Modular form 20160.2.a.ca
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
