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SageMath
E = EllipticCurve("de1")
E.isogeny_class()
Elliptic curves in class 55440.de
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 55440.de1 | 55440bb4 | \([0, 0, 0, -146652267, -605423261126]\) | \(233632133015204766393938/29145526885986328125\) | \(43514038476562500000000000\) | \([2]\) | \(15728640\) | \(3.6492\) | |
| 55440.de2 | 55440bb2 | \([0, 0, 0, -36631587, 75538735666]\) | \(7282213870869695463556/912102595400390625\) | \(680880939056010000000000\) | \([2, 2]\) | \(7864320\) | \(3.3027\) | |
| 55440.de3 | 55440bb1 | \([0, 0, 0, -35450607, 81241215694]\) | \(26401417552259125806544/507547744790625\) | \(94720590323805600000\) | \([2]\) | \(3932160\) | \(2.9561\) | \(\Gamma_0(N)\)-optimal |
| 55440.de4 | 55440bb3 | \([0, 0, 0, 54493413, 391542010666]\) | \(11986661998777424518222/51295853620928503125\) | \(-76584299089217287737600000\) | \([2]\) | \(15728640\) | \(3.6492\) |
Rank
sage: E.rank()
The elliptic curves in class 55440.de have rank \(1\).
Complex multiplication
The elliptic curves in class 55440.de do not have complex multiplication.Modular form 55440.2.a.de
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
