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SageMath
E = EllipticCurve("dc1")
E.isogeny_class()
Elliptic curves in class 72450.dc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 72450.dc1 | 72450dt4 | \([1, -1, 1, -8307022505, -291405429080503]\) | \(5565604209893236690185614401/229307220930246900000\) | \(2611952563408593595312500000\) | \([2]\) | \(117964800\) | \(4.3422\) | |
| 72450.dc2 | 72450dt3 | \([1, -1, 1, -2532550505, 45238720599497]\) | \(157706830105239346386477121/13650704956054687500000\) | \(155490061140060424804687500000\) | \([2]\) | \(117964800\) | \(4.3422\) | |
| 72450.dc3 | 72450dt2 | \([1, -1, 1, -544522505, -4084254080503]\) | \(1567558142704512417614401/274462175610000000000\) | \(3126295719057656250000000000\) | \([2, 2]\) | \(58982400\) | \(3.9956\) | |
| 72450.dc4 | 72450dt1 | \([1, -1, 1, 64885495, -365646464503]\) | \(2652277923951208297919/6605028468326400000\) | \(-75235402397030400000000000\) | \([4]\) | \(29491200\) | \(3.6491\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72450.dc have rank \(0\).
Complex multiplication
The elliptic curves in class 72450.dc do not have complex multiplication.Modular form 72450.2.a.dc
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.
