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SageMath
E = EllipticCurve("eu1")
E.isogeny_class()
Elliptic curves in class 212160.eu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 212160.eu1 | 212160fq4 | \([0, 1, 0, -29121, -1917345]\) | \(83350372210568/258984375\) | \(8486400000000\) | \([2]\) | \(524288\) | \(1.3485\) | |
| 212160.eu2 | 212160fq2 | \([0, 1, 0, -2601, -2601]\) | \(475282454464/274730625\) | \(1125296640000\) | \([2, 2]\) | \(262144\) | \(1.0019\) | |
| 212160.eu3 | 212160fq1 | \([0, 1, 0, -1756, 27650]\) | \(9361912326976/36415275\) | \(2330577600\) | \([2]\) | \(131072\) | \(0.65534\) | \(\Gamma_0(N)\)-optimal |
| 212160.eu4 | 212160fq3 | \([0, 1, 0, 10399, -10401]\) | \(3794956027192/2198690325\) | \(-72046684569600\) | \([2]\) | \(524288\) | \(1.3485\) |
Rank
sage: E.rank()
The elliptic curves in class 212160.eu have rank \(1\).
Complex multiplication
The elliptic curves in class 212160.eu do not have complex multiplication.Modular form 212160.2.a.eu
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.
