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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 36414cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 36414.cg5 | 36414cf1 | \([1, -1, 1, -1355, -38505]\) | \(-15625/28\) | \(-492696058428\) | \([2]\) | \(36864\) | \(0.93383\) | \(\Gamma_0(N)\)-optimal |
| 36414.cg4 | 36414cf2 | \([1, -1, 1, -27365, -1734357]\) | \(128787625/98\) | \(1724436204498\) | \([2]\) | \(73728\) | \(1.2804\) | |
| 36414.cg6 | 36414cf3 | \([1, -1, 1, 11650, 809421]\) | \(9938375/21952\) | \(-386273709807552\) | \([2]\) | \(110592\) | \(1.4831\) | |
| 36414.cg3 | 36414cf4 | \([1, -1, 1, -92390, 8882925]\) | \(4956477625/941192\) | \(16561485307998792\) | \([2]\) | \(221184\) | \(1.8297\) | |
| 36414.cg2 | 36414cf5 | \([1, -1, 1, -443525, 114129789]\) | \(-548347731625/1835008\) | \(-32289328885137408\) | \([2]\) | \(331776\) | \(2.0324\) | |
| 36414.cg1 | 36414cf6 | \([1, -1, 1, -7102085, 7286730621]\) | \(2251439055699625/25088\) | \(441455668351488\) | \([2]\) | \(663552\) | \(2.3790\) |
Rank
sage: E.rank()
The elliptic curves in class 36414cf have rank \(1\).
Complex multiplication
The elliptic curves in class 36414cf do not have complex multiplication.Modular form 36414.2.a.cf
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
