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SageMath
E = EllipticCurve("eh1")
E.isogeny_class()
Elliptic curves in class 31680.eh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 31680.eh1 | 31680ee4 | \([0, 0, 0, -255612, -49741616]\) | \(154639330142416/33275\) | \(397434470400\) | \([2]\) | \(165888\) | \(1.6103\) | |
| 31680.eh2 | 31680ee3 | \([0, 0, 0, -16032, -771464]\) | \(610462990336/8857805\) | \(6612316001280\) | \([2]\) | \(82944\) | \(1.2638\) | |
| 31680.eh3 | 31680ee2 | \([0, 0, 0, -3612, -47216]\) | \(436334416/171875\) | \(2052864000000\) | \([2]\) | \(55296\) | \(1.0610\) | |
| 31680.eh4 | 31680ee1 | \([0, 0, 0, -1632, 24856]\) | \(643956736/15125\) | \(11290752000\) | \([2]\) | \(27648\) | \(0.71445\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 31680.eh have rank \(1\).
Complex multiplication
The elliptic curves in class 31680.eh do not have complex multiplication.Modular form 31680.2.a.eh
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
