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SageMath
E = EllipticCurve("dc1")
E.isogeny_class()
Elliptic curves in class 78400.dc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 78400.dc1 | 78400kp2 | \([0, -1, 0, -393633, -94928063]\) | \(-349938025/8\) | \(-154204897280000\) | \([]\) | \(414720\) | \(1.8359\) | |
| 78400.dc2 | 78400kp3 | \([0, -1, 0, -236833, 53561537]\) | \(-121945/32\) | \(-385512243200000000\) | \([]\) | \(691200\) | \(2.0913\) | |
| 78400.dc3 | 78400kp1 | \([0, -1, 0, -1633, -299263]\) | \(-25/2\) | \(-38551224320000\) | \([]\) | \(138240\) | \(1.2866\) | \(\Gamma_0(N)\)-optimal |
| 78400.dc4 | 78400kp4 | \([0, -1, 0, 1723167, -395278463]\) | \(46969655/32768\) | \(-394764537036800000000\) | \([]\) | \(2073600\) | \(2.6406\) |
Rank
sage: E.rank()
The elliptic curves in class 78400.dc have rank \(2\).
Complex multiplication
The elliptic curves in class 78400.dc do not have complex multiplication.Modular form 78400.2.a.dc
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.
\(\left(\begin{array}{rrrr} 1 & 15 & 3 & 5 \\ 15 & 1 & 5 & 3 \\ 3 & 5 & 1 & 15 \\ 5 & 3 & 15 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
