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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 62475cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 62475.cq5 | 62475cf1 | \([1, 0, 1, 42849, -8446427]\) | \(4733169839/19518975\) | \(-35881060777734375\) | \([2]\) | \(589824\) | \(1.8592\) | \(\Gamma_0(N)\)-optimal |
| 62475.cq4 | 62475cf2 | \([1, 0, 1, -453276, -103702427]\) | \(5602762882081/716900625\) | \(1317853775478515625\) | \([2, 2]\) | \(1179648\) | \(2.2058\) | |
| 62475.cq3 | 62475cf3 | \([1, 0, 1, -1831401, 847203823]\) | \(369543396484081/45120132225\) | \(82942788064672265625\) | \([2, 2]\) | \(2359296\) | \(2.5523\) | |
| 62475.cq2 | 62475cf4 | \([1, 0, 1, -7013151, -7149008177]\) | \(20751759537944401/418359375\) | \(769055657958984375\) | \([2]\) | \(2359296\) | \(2.5523\) | |
| 62475.cq6 | 62475cf5 | \([1, 0, 1, 2670474, 4358666323]\) | \(1145725929069119/5127181719135\) | \(-9425121907414275234375\) | \([2]\) | \(4718592\) | \(2.8989\) | |
| 62475.cq1 | 62475cf6 | \([1, 0, 1, -28383276, 58199253823]\) | \(1375634265228629281/24990412335\) | \(45939015950006484375\) | \([2]\) | \(4718592\) | \(2.8989\) |
Rank
sage: E.rank()
The elliptic curves in class 62475cf have rank \(0\).
Complex multiplication
The elliptic curves in class 62475cf do not have complex multiplication.Modular form 62475.2.a.cf
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.
