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SageMath
E = EllipticCurve("eb1")
E.isogeny_class()
Elliptic curves in class 55440eb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55440.cw3 | 55440eb1 | \([0, 0, 0, -29307, 1929706]\) | \(932288503609/779625\) | \(2327947776000\) | \([2]\) | \(147456\) | \(1.3009\) | \(\Gamma_0(N)\)-optimal |
| 55440.cw2 | 55440eb2 | \([0, 0, 0, -35787, 1013434]\) | \(1697509118089/833765625\) | \(2489610816000000\) | \([2, 2]\) | \(294912\) | \(1.6475\) | |
| 55440.cw4 | 55440eb3 | \([0, 0, 0, 130533, 7766026]\) | \(82375335041831/56396484375\) | \(-168399000000000000\) | \([4]\) | \(589824\) | \(1.9940\) | |
| 55440.cw1 | 55440eb4 | \([0, 0, 0, -305787, -64380566]\) | \(1058993490188089/13182390375\) | \(39362406741504000\) | \([2]\) | \(589824\) | \(1.9940\) |
Rank
sage: E.rank()
The elliptic curves in class 55440eb have rank \(2\).
Complex multiplication
The elliptic curves in class 55440eb do not have complex multiplication.Modular form 55440.2.a.eb
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.
