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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 201810.cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 201810.cr1 | 201810a5 | \([1, 0, 0, -16144820, 24967469250]\) | \(524388516989299201/3150\) | \(2795636595150\) | \([2]\) | \(7372800\) | \(2.4538\) | |
| 201810.cr2 | 201810a3 | \([1, 0, 0, -1009070, 390038400]\) | \(128031684631201/9922500\) | \(8806255274722500\) | \([2, 2]\) | \(3686400\) | \(2.1072\) | |
| 201810.cr3 | 201810a6 | \([1, 0, 0, -941800, 444298382]\) | \(-104094944089921/35880468750\) | \(-31844048091630468750\) | \([2]\) | \(7372800\) | \(2.4538\) | |
| 201810.cr4 | 201810a4 | \([1, 0, 0, -355590, -77169048]\) | \(5602762882081/345888060\) | \(306976926463948860\) | \([2]\) | \(3686400\) | \(2.1072\) | |
| 201810.cr5 | 201810a2 | \([1, 0, 0, -67290, 5227092]\) | \(37966934881/8643600\) | \(7671226817091600\) | \([2, 2]\) | \(1843200\) | \(1.7606\) | |
| 201810.cr6 | 201810a1 | \([1, 0, 0, 9590, 506660]\) | \(109902239/188160\) | \(-166992692616960\) | \([2]\) | \(921600\) | \(1.4140\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 201810.cr have rank \(0\).
Complex multiplication
The elliptic curves in class 201810.cr do not have complex multiplication.Modular form 201810.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
