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SageMath
E = EllipticCurve("et1")
E.isogeny_class()
Elliptic curves in class 212160.et
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.et1 | 212160bm4 | \([0, 1, 0, -1543361, -738498561]\) | \(12407281149601054088/83573056125\) | \(2738521903104000\) | \([2]\) | \(2260992\) | \(2.1439\) | |
| 212160.et2 | 212160bm2 | \([0, 1, 0, -98361, -11085561]\) | \(25694311545152704/1984928765625\) | \(8130268224000000\) | \([2, 2]\) | \(1130496\) | \(1.7973\) | |
| 212160.et3 | 212160bm1 | \([0, 1, 0, -20236, 898814]\) | \(14319874494773056/2751708984375\) | \(176109375000000\) | \([2]\) | \(565248\) | \(1.4507\) | \(\Gamma_0(N)\)-optimal |
| 212160.et4 | 212160bm3 | \([0, 1, 0, 96639, -49422561]\) | \(3045976617265912/34006817524875\) | \(-1114335396655104000\) | \([2]\) | \(2260992\) | \(2.1439\) |
Rank
sage: E.rank()
The elliptic curves in class 212160.et have rank \(1\).
Complex multiplication
The elliptic curves in class 212160.et do not have complex multiplication.Modular form 212160.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
