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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 110670.cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 110670.cm1 | 110670cp8 | \([1, 0, 0, -258731208280, -50654853221670100]\) | \(1915447099311696788795300773853656437121/2359330710565906457438184052500\) | \(2359330710565906457438184052500\) | \([2]\) | \(788529152\) | \(5.1081\) | |
| 110670.cm2 | 110670cp4 | \([1, 0, 0, -36426360100, 2675908664482832]\) | \(5345287166085790635663218704920974401/226978257155929261920000\) | \(226978257155929261920000\) | \([8]\) | \(197132288\) | \(4.4149\) | |
| 110670.cm3 | 110670cp6 | \([1, 0, 0, -16306945780, -777467273017600]\) | \(479558500651862026155270257138637121/16398477508430925116750756250000\) | \(16398477508430925116750756250000\) | \([2, 2]\) | \(394264576\) | \(4.7615\) | |
| 110670.cm4 | 110670cp3 | \([1, 0, 0, -2525695780, 32101233232400]\) | \(1781832302709884421209712638637121/585975791707934414062500000000\) | \(585975791707934414062500000000\) | \([2, 4]\) | \(197132288\) | \(4.4149\) | |
| 110670.cm5 | 110670cp2 | \([1, 0, 0, -2276760100, 41806588162832]\) | \(1305195379419707692723460338574401/268915325631261926400000000\) | \(268915325631261926400000000\) | \([2, 8]\) | \(98566144\) | \(4.0683\) | |
| 110670.cm6 | 110670cp1 | \([1, 0, 0, -126851620, 800523760400]\) | \(-225741686871429146260559062081/146211902909299839467520000\) | \(-146211902909299839467520000\) | \([8]\) | \(49283072\) | \(3.7218\) | \(\Gamma_0(N)\)-optimal |
| 110670.cm7 | 110670cp7 | \([1, 0, 0, 5617316720, -2712410599365100]\) | \(19602454118850102896203267379162879/3189643093950949672061474484052500\) | \(-3189643093950949672061474484052500\) | \([2]\) | \(788529152\) | \(5.1081\) | |
| 110670.cm8 | 110670cp5 | \([1, 0, 0, 7272583340, 220528019677472]\) | \(42539250356844378683161410804461759/45626792684197425842285156250000\) | \(-45626792684197425842285156250000\) | \([4]\) | \(394264576\) | \(4.7615\) |
Rank
sage: E.rank()
The elliptic curves in class 110670.cm have rank \(0\).
Complex multiplication
The elliptic curves in class 110670.cm do not have complex multiplication.Modular form 110670.2.a.cm
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 2 & 4 & 8 & 16 & 4 & 8 \\ 16 & 1 & 8 & 4 & 2 & 4 & 16 & 8 \\ 2 & 8 & 1 & 2 & 4 & 8 & 2 & 4 \\ 4 & 4 & 2 & 1 & 2 & 4 & 4 & 2 \\ 8 & 2 & 4 & 2 & 1 & 2 & 8 & 4 \\ 16 & 4 & 8 & 4 & 2 & 1 & 16 & 8 \\ 4 & 16 & 2 & 4 & 8 & 16 & 1 & 8 \\ 8 & 8 & 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
