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SageMath
E = EllipticCurve("de1")
E.isogeny_class()
Elliptic curves in class 50400de
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 50400.bx3 | 50400de1 | \([0, 0, 0, -1181325, -494093500]\) | \(250094631024064/62015625\) | \(45209390625000000\) | \([2, 2]\) | \(589824\) | \(2.1834\) | \(\Gamma_0(N)\)-optimal |
| 50400.bx4 | 50400de2 | \([0, 0, 0, -1040700, -616156000]\) | \(-2671731885376/1969120125\) | \(-91871268552000000000\) | \([2]\) | \(1179648\) | \(2.5300\) | |
| 50400.bx2 | 50400de3 | \([0, 0, 0, -1323075, -368077750]\) | \(43919722445768/15380859375\) | \(89701171875000000000\) | \([2]\) | \(1179648\) | \(2.5300\) | |
| 50400.bx1 | 50400de4 | \([0, 0, 0, -18900075, -31625937250]\) | \(128025588102048008/7875\) | \(45927000000000\) | \([2]\) | \(1179648\) | \(2.5300\) |
Rank
sage: E.rank()
The elliptic curves in class 50400de have rank \(1\).
Complex multiplication
The elliptic curves in class 50400de do not have complex multiplication.Modular form 50400.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 2 & 2 \\ 2 & 1 & 4 & 4 \\ 2 & 4 & 1 & 4 \\ 2 & 4 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
