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SageMath
E = EllipticCurve("fa1")
E.isogeny_class()
Elliptic curves in class 270480fa
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 270480.fa4 | 270480fa1 | \([0, 1, 0, -2672779496, -46372643606220]\) | \(4381924769947287308715481/608122186185572352000\) | \(293048185170110061119275008000\) | \([2]\) | \(371589120\) | \(4.3807\) | \(\Gamma_0(N)\)-optimal |
| 270480.fa2 | 270480fa2 | \([0, 1, 0, -41224003816, -3221559422029516]\) | \(16077778198622525072705635801/388799208512064000000\) | \(187358568784837908627456000000\) | \([2, 2]\) | \(743178240\) | \(4.7273\) | |
| 270480.fa3 | 270480fa3 | \([0, 1, 0, -39687363816, -3472799447373516]\) | \(-14346048055032350809895395801/2509530875136386550792000\) | \(-1209318596316859356422668320768000\) | \([2]\) | \(1486356480\) | \(5.0738\) | |
| 270480.fa1 | 270480fa4 | \([0, 1, 0, -659580232936, -206181656396187340]\) | \(65853432878493908038433301506521/38511703125000000\) | \(18558416326464000000000000\) | \([2]\) | \(1486356480\) | \(5.0738\) |
Rank
sage: E.rank()
The elliptic curves in class 270480fa have rank \(0\).
Complex multiplication
The elliptic curves in class 270480fa do not have complex multiplication.Modular form 270480.2.a.fa
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.
