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SageMath
E = EllipticCurve("hw1")
E.isogeny_class()
Elliptic curves in class 202800hw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 202800.ht3 | 202800hw1 | \([0, 1, 0, -30983, -1220712]\) | \(2725888/1053\) | \(1270657469250000\) | \([2]\) | \(688128\) | \(1.5972\) | \(\Gamma_0(N)\)-optimal |
| 202800.ht2 | 202800hw2 | \([0, 1, 0, -221108, 39085788]\) | \(61918288/1521\) | \(29366305956000000\) | \([2, 2]\) | \(1376256\) | \(1.9437\) | |
| 202800.ht1 | 202800hw3 | \([0, 1, 0, -3516608, 2537074788]\) | \(62275269892/39\) | \(3011928816000000\) | \([2]\) | \(2752512\) | \(2.2903\) | |
| 202800.ht4 | 202800hw4 | \([0, 1, 0, 32392, 123754788]\) | \(48668/85683\) | \(-6617207608752000000\) | \([2]\) | \(2752512\) | \(2.2903\) |
Rank
sage: E.rank()
The elliptic curves in class 202800hw have rank \(0\).
Complex multiplication
The elliptic curves in class 202800hw do not have complex multiplication.Modular form 202800.2.a.hw
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.
