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SageMath
E = EllipticCurve("cn1")
E.isogeny_class()
Elliptic curves in class 198900.cn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 198900.cn1 | 198900cc3 | \([0, 0, 0, -1114500, 61131125]\) | \(840033089536000/477272151837\) | \(86982849672293250000\) | \([2]\) | \(4976640\) | \(2.5159\) | |
| 198900.cn2 | 198900cc1 | \([0, 0, 0, -709500, -230023375]\) | \(216727177216000/2738853\) | \(499155959250000\) | \([2]\) | \(1658880\) | \(1.9666\) | \(\Gamma_0(N)\)-optimal |
| 198900.cn3 | 198900cc2 | \([0, 0, 0, -690375, -243009250]\) | \(-12479332642000/1526829993\) | \(-4452236259588000000\) | \([2]\) | \(3317760\) | \(2.3132\) | |
| 198900.cn4 | 198900cc4 | \([0, 0, 0, 4412625, 486719750]\) | \(3258571509326000/1920843121977\) | \(-5601178543684932000000\) | \([2]\) | \(9953280\) | \(2.8625\) |
Rank
sage: E.rank()
The elliptic curves in class 198900.cn have rank \(1\).
Complex multiplication
The elliptic curves in class 198900.cn do not have complex multiplication.Modular form 198900.2.a.cn
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
