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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 188760.cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 188760.cd1 | 188760b3 | \([0, 1, 0, -67953640, 215586507200]\) | \(19129597231400697604/26325\) | \(47755615564800\) | \([2]\) | \(8847360\) | \(2.7910\) | |
| 188760.cd2 | 188760b2 | \([0, 1, 0, -4247140, 3367414400]\) | \(18681746265374416/693005625\) | \(314291644935840000\) | \([2, 2]\) | \(4423680\) | \(2.4445\) | |
| 188760.cd3 | 188760b4 | \([0, 1, 0, -4051120, 3692493968]\) | \(-4053153720264484/903687890625\) | \(-1639360740560400000000\) | \([2]\) | \(8847360\) | \(2.7910\) | |
| 188760.cd4 | 188760b1 | \([0, 1, 0, -277735, 47404058]\) | \(83587439220736/13990184325\) | \(396551438927701200\) | \([2]\) | \(2211840\) | \(2.0979\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 188760.cd have rank \(1\).
Complex multiplication
The elliptic curves in class 188760.cd do not have complex multiplication.Modular form 188760.2.a.cd
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.
