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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 330330.cr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 330330.cr1 | 330330cr7 | \([1, 0, 1, -4863045058, 130529659590878]\) | \(7179471593960193209684686321/49441793310\) | \(87589152798056910\) | \([2]\) | \(159252480\) | \(3.7867\) | |
| 330330.cr2 | 330330cr6 | \([1, 0, 1, -303940508, 2039504237318]\) | \(1752803993935029634719121/4599740941532100\) | \(8148721662121548608100\) | \([2, 2]\) | \(79626240\) | \(3.4402\) | |
| 330330.cr3 | 330330cr8 | \([1, 0, 1, -300205238, 2092075921406]\) | \(-1688971789881664420008241/89901485966373558750\) | \(-159265966380074708112708750\) | \([2]\) | \(159252480\) | \(3.7867\) | |
| 330330.cr4 | 330330cr4 | \([1, 0, 1, -60065008, 178876879718]\) | \(13527956825588849127121/25701087819771000\) | \(45531044839081332531000\) | \([2]\) | \(53084160\) | \(3.2374\) | |
| 330330.cr5 | 330330cr3 | \([1, 0, 1, -19229928, 31041921766]\) | \(443915739051786565201/21894701746029840\) | \(38787799719898369380240\) | \([2]\) | \(39813120\) | \(3.0936\) | |
| 330330.cr6 | 330330cr2 | \([1, 0, 1, -5010008, 762943718]\) | \(7850236389974007121/4400862921000000\) | \(7796397117189681000000\) | \([2, 2]\) | \(26542080\) | \(2.8909\) | |
| 330330.cr7 | 330330cr1 | \([1, 0, 1, -3112728, -2102707994]\) | \(1882742462388824401/11650189824000\) | \(20639021934795264000\) | \([2]\) | \(13271040\) | \(2.5443\) | \(\Gamma_0(N)\)-optimal |
| 330330.cr8 | 330330cr5 | \([1, 0, 1, 19688512, 6058306406]\) | \(476437916651992691759/284661685546875000\) | \(-504295540309107421875000\) | \([2]\) | \(53084160\) | \(3.2374\) |
Rank
sage: E.rank()
The elliptic curves in class 330330.cr have rank \(2\).
Complex multiplication
The elliptic curves in class 330330.cr do not have complex multiplication.Modular form 330330.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
