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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 23232ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 23232.dj3 | 23232ca1 | \([0, 1, 0, -50497, 4314143]\) | \(30664297/297\) | \(137928013774848\) | \([2]\) | \(92160\) | \(1.5329\) | \(\Gamma_0(N)\)-optimal |
| 23232.dj2 | 23232ca2 | \([0, 1, 0, -89217, -3251745]\) | \(169112377/88209\) | \(40964620091129856\) | \([2, 2]\) | \(184320\) | \(1.8795\) | |
| 23232.dj4 | 23232ca3 | \([0, 1, 0, 336703, -24973665]\) | \(9090072503/5845851\) | \(-2714837095130333184\) | \([2]\) | \(368640\) | \(2.2260\) | |
| 23232.dj1 | 23232ca4 | \([0, 1, 0, -1134657, -465127137]\) | \(347873904937/395307\) | \(183582186334322688\) | \([2]\) | \(368640\) | \(2.2260\) |
Rank
sage: E.rank()
The elliptic curves in class 23232ca have rank \(1\).
Complex multiplication
The elliptic curves in class 23232ca do not have complex multiplication.Modular form 23232.2.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 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.
