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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 101430cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 101430.cs3 | 101430cr1 | \([1, -1, 0, -211689, 2134157773]\) | \(-12232183057921/22933241856000\) | \(-1966895195943960576000\) | \([2]\) | \(7962624\) | \(2.7649\) | \(\Gamma_0(N)\)-optimal |
| 101430.cs2 | 101430cr2 | \([1, -1, 0, -26177769, 50934808525]\) | \(23131609187144855041/322060536000000\) | \(27621882899900856000000\) | \([2]\) | \(15925248\) | \(3.1114\) | |
| 101430.cs4 | 101430cr3 | \([1, -1, 0, 1905111, -57605748467]\) | \(8915971454369279/16719623332762560\) | \(-1433977237832136985229760\) | \([2]\) | \(23887872\) | \(3.3142\) | |
| 101430.cs1 | 101430cr4 | \([1, -1, 0, -212720769, -1168638078875]\) | \(12411881707829361287041/303132494474220600\) | \(25998498200107835360292600\) | \([2]\) | \(47775744\) | \(3.6607\) |
Rank
sage: E.rank()
The elliptic curves in class 101430cr have rank \(1\).
Complex multiplication
The elliptic curves in class 101430cr do not have complex multiplication.Modular form 101430.2.a.cr
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.
