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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 25200bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25200.cs3 | 25200bd1 | \([0, 0, 0, -71175, -7308250]\) | \(13674725584/945\) | \(2755620000000\) | \([2]\) | \(73728\) | \(1.4408\) | \(\Gamma_0(N)\)-optimal |
| 25200.cs2 | 25200bd2 | \([0, 0, 0, -75675, -6331750]\) | \(4108974916/893025\) | \(10416243600000000\) | \([2, 2]\) | \(147456\) | \(1.7874\) | |
| 25200.cs4 | 25200bd3 | \([0, 0, 0, 167325, -38650750]\) | \(22208984782/40516875\) | \(-945177660000000000\) | \([2]\) | \(294912\) | \(2.1339\) | |
| 25200.cs1 | 25200bd4 | \([0, 0, 0, -390675, 88483250]\) | \(282678688658/18600435\) | \(433910947680000000\) | \([2]\) | \(294912\) | \(2.1339\) |
Rank
sage: E.rank()
The elliptic curves in class 25200bd have rank \(0\).
Complex multiplication
The elliptic curves in class 25200bd do not have complex multiplication.Modular form 25200.2.a.bd
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.
