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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 5850b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5850.z4 | 5850b1 | \([1, -1, 0, 2433, -36659]\) | \(3774555693/3515200\) | \(-1482975000000\) | \([2]\) | \(13824\) | \(1.0226\) | \(\Gamma_0(N)\)-optimal |
| 5850.z3 | 5850b2 | \([1, -1, 0, -12567, -321659]\) | \(520300455507/193072360\) | \(81452401875000\) | \([2]\) | \(27648\) | \(1.3692\) | |
| 5850.z2 | 5850b3 | \([1, -1, 0, -56067, -5152159]\) | \(-63378025803/812500\) | \(-249881835937500\) | \([2]\) | \(41472\) | \(1.5720\) | |
| 5850.z1 | 5850b4 | \([1, -1, 0, -899817, -328308409]\) | \(261984288445803/42250\) | \(12993855468750\) | \([2]\) | \(82944\) | \(1.9185\) |
Rank
sage: E.rank()
The elliptic curves in class 5850b have rank \(1\).
Complex multiplication
The elliptic curves in class 5850b do not have complex multiplication.Modular form 5850.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
