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SageMath
E = EllipticCurve("fe1")
E.isogeny_class()
Elliptic curves in class 109200fe
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 109200.fd3 | 109200fe1 | \([0, 1, 0, -22608, 1264788]\) | \(19968681097/628992\) | \(40255488000000\) | \([2]\) | \(294912\) | \(1.3854\) | \(\Gamma_0(N)\)-optimal |
| 109200.fd2 | 109200fe2 | \([0, 1, 0, -54608, -3151212]\) | \(281397674377/96589584\) | \(6181733376000000\) | \([2, 2]\) | \(589824\) | \(1.7319\) | |
| 109200.fd4 | 109200fe3 | \([0, 1, 0, 161392, -21727212]\) | \(7264187703863/7406095788\) | \(-473990130432000000\) | \([2]\) | \(1179648\) | \(2.0785\) | |
| 109200.fd1 | 109200fe4 | \([0, 1, 0, -782608, -266687212]\) | \(828279937799497/193444524\) | \(12380449536000000\) | \([2]\) | \(1179648\) | \(2.0785\) |
Rank
sage: E.rank()
The elliptic curves in class 109200fe have rank \(1\).
Complex multiplication
The elliptic curves in class 109200fe do not have complex multiplication.Modular form 109200.2.a.fe
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 Cremona 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 Cremona labels.
