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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 9900.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 9900.e1 | 9900l4 | \([0, 0, 0, -56249175, 162376190750]\) | \(6749703004355978704/5671875\) | \(16539187500000000\) | \([2]\) | \(331776\) | \(2.8476\) | |
| 9900.e2 | 9900l3 | \([0, 0, 0, -3514800, 2538300125]\) | \(-26348629355659264/24169921875\) | \(-4404968261718750000\) | \([2]\) | \(165888\) | \(2.5010\) | |
| 9900.e3 | 9900l2 | \([0, 0, 0, -710175, 212111750]\) | \(13584145739344/1195803675\) | \(3486963516300000000\) | \([2]\) | \(110592\) | \(2.2983\) | |
| 9900.e4 | 9900l1 | \([0, 0, 0, 49200, 15433625]\) | \(72268906496/606436875\) | \(-110523120468750000\) | \([2]\) | \(55296\) | \(1.9517\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 9900.e have rank \(1\).
Complex multiplication
The elliptic curves in class 9900.e do not have complex multiplication.Modular form 9900.2.a.e
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.
