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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 18240ck
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 18240.cf4 | 18240ck1 | \([0, 1, 0, 799, 7839]\) | \(214921799/218880\) | \(-57378078720\) | \([2]\) | \(24576\) | \(0.75126\) | \(\Gamma_0(N)\)-optimal |
| 18240.cf3 | 18240ck2 | \([0, 1, 0, -4321, 68255]\) | \(34043726521/11696400\) | \(3066141081600\) | \([2, 2]\) | \(49152\) | \(1.0978\) | |
| 18240.cf2 | 18240ck3 | \([0, 1, 0, -28641, -1823841]\) | \(9912050027641/311647500\) | \(81696522240000\) | \([2]\) | \(98304\) | \(1.4444\) | |
| 18240.cf1 | 18240ck4 | \([0, 1, 0, -61921, 5908895]\) | \(100162392144121/23457780\) | \(6149316280320\) | \([2]\) | \(98304\) | \(1.4444\) |
Rank
sage: E.rank()
The elliptic curves in class 18240ck have rank \(1\).
Complex multiplication
The elliptic curves in class 18240ck do not have complex multiplication.Modular form 18240.2.a.ck
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.
