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SageMath
E = EllipticCurve("hd1")
E.isogeny_class()
Elliptic curves in class 369600.hd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 369600.hd1 | 369600hd4 | \([0, -1, 0, -425905793, 3383174665857]\) | \(260744057755293612689909/8504954620259328\) | \(278690352996657659904000\) | \([2]\) | \(73728000\) | \(3.5927\) | |
| 369600.hd2 | 369600hd3 | \([0, -1, 0, -27774593, 48029603457]\) | \(72313087342699809269/11447096545640448\) | \(375098459607546200064000\) | \([2]\) | \(36864000\) | \(3.2462\) | |
| 369600.hd3 | 369600hd2 | \([0, -1, 0, -7536193, -7887505343]\) | \(1444540994277943589/15251205665388\) | \(499751507243433984000\) | \([2]\) | \(14745600\) | \(2.7880\) | |
| 369600.hd4 | 369600hd1 | \([0, -1, 0, -7516993, -7930071743]\) | \(1433528304665250149/162339408\) | \(5319537721344000\) | \([2]\) | \(7372800\) | \(2.4414\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 369600.hd have rank \(1\).
Complex multiplication
The elliptic curves in class 369600.hd do not have complex multiplication.Modular form 369600.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
