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SageMath
E = EllipticCurve("hd1")
E.isogeny_class()
Elliptic curves in class 187200hd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 187200.bg4 | 187200hd1 | \([0, 0, 0, 155700, 18458000]\) | \(3774555693/3515200\) | \(-388752998400000000\) | \([2]\) | \(2654208\) | \(2.0624\) | \(\Gamma_0(N)\)-optimal |
| 187200.bg3 | 187200hd2 | \([0, 0, 0, -804300, 166298000]\) | \(520300455507/193072360\) | \(21352258437120000000\) | \([2]\) | \(5308416\) | \(2.4089\) | |
| 187200.bg2 | 187200hd3 | \([0, 0, 0, -3588300, 2645082000]\) | \(-63378025803/812500\) | \(-65505024000000000000\) | \([2]\) | \(7962624\) | \(2.6117\) | |
| 187200.bg1 | 187200hd4 | \([0, 0, 0, -57588300, 168209082000]\) | \(261984288445803/42250\) | \(3406261248000000000\) | \([2]\) | \(15925248\) | \(2.9582\) |
Rank
sage: E.rank()
The elliptic curves in class 187200hd have rank \(1\).
Complex multiplication
The elliptic curves in class 187200hd do not have complex multiplication.Modular form 187200.2.a.hd
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
