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SageMath
E = EllipticCurve("fl1")
E.isogeny_class()
Elliptic curves in class 277200fl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 277200.fl4 | 277200fl1 | \([0, 0, 0, -954075, 358690250]\) | \(2058561081361/12705\) | \(592764480000000\) | \([2]\) | \(3145728\) | \(2.0213\) | \(\Gamma_0(N)\)-optimal |
| 277200.fl3 | 277200fl2 | \([0, 0, 0, -972075, 344452250]\) | \(2177286259681/161417025\) | \(7531072718400000000\) | \([2, 2]\) | \(6291456\) | \(2.3679\) | |
| 277200.fl5 | 277200fl3 | \([0, 0, 0, 917925, 1521922250]\) | \(1833318007919/22507682505\) | \(-1050118434953280000000\) | \([2]\) | \(12582912\) | \(2.7145\) | |
| 277200.fl2 | 277200fl4 | \([0, 0, 0, -3150075, -1744249750]\) | \(74093292126001/14707625625\) | \(686198981160000000000\) | \([2, 2]\) | \(12582912\) | \(2.7145\) | |
| 277200.fl6 | 277200fl5 | \([0, 0, 0, 6551925, -10369327750]\) | \(666688497209279/1381398046875\) | \(-64450507275000000000000\) | \([2]\) | \(25165824\) | \(3.0610\) | |
| 277200.fl1 | 277200fl6 | \([0, 0, 0, -47700075, -126796099750]\) | \(257260669489908001/14267882475\) | \(665682324753600000000\) | \([2]\) | \(25165824\) | \(3.0610\) |
Rank
sage: E.rank()
The elliptic curves in class 277200fl have rank \(2\).
Complex multiplication
The elliptic curves in class 277200fl do not have complex multiplication.Modular form 277200.2.a.fl
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
