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SageMath
E = EllipticCurve("et1")
E.isogeny_class()
Elliptic curves in class 161700.et
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 161700.et1 | 161700bi4 | \([0, 1, 0, -306245508, 2062676933988]\) | \(6749703004355978704/5671875\) | \(2669161687500000000\) | \([2]\) | \(14929920\) | \(3.2712\) | |
| 161700.et2 | 161700bi3 | \([0, 1, 0, -19136133, 32239433988]\) | \(-26348629355659264/24169921875\) | \(-710891784667968750000\) | \([2]\) | \(7464960\) | \(2.9247\) | |
| 161700.et3 | 161700bi2 | \([0, 1, 0, -3866508, 2693315988]\) | \(13584145739344/1195803675\) | \(562740426240300000000\) | \([2]\) | \(4976640\) | \(2.7219\) | |
| 161700.et4 | 161700bi1 | \([0, 1, 0, 267867, 196153488]\) | \(72268906496/606436875\) | \(-17836672976718750000\) | \([2]\) | \(2488320\) | \(2.3753\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 161700.et have rank \(1\).
Complex multiplication
The elliptic curves in class 161700.et do not have complex multiplication.Modular form 161700.2.a.et
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.
