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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 221760.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221760.k1 | 221760fg4 | \([0, 0, 0, -11383788, -14783535632]\) | \(6829778934856027592/2401245\) | \(57360633200640\) | \([2]\) | \(5242880\) | \(2.4328\) | |
| 221760.k2 | 221760fg2 | \([0, 0, 0, -711588, -230923712]\) | \(13345107693305536/7909434225\) | \(23617444044902400\) | \([2, 2]\) | \(2621440\) | \(2.0862\) | |
| 221760.k3 | 221760fg3 | \([0, 0, 0, -580908, -318374768]\) | \(-907545319055432/1307889226875\) | \(-31242690441768960000\) | \([2]\) | \(5242880\) | \(2.4328\) | |
| 221760.k4 | 221760fg1 | \([0, 0, 0, -52743, -2172728]\) | \(347784878972224/157553777535\) | \(7350829044672960\) | \([2]\) | \(1310720\) | \(1.7396\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 221760.k have rank \(1\).
Complex multiplication
The elliptic curves in class 221760.k do not have complex multiplication.Modular form 221760.2.a.k
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.
