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SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 110670.co
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 110670.co1 | 110670cm8 | \([1, 0, 0, -1096991420, 13971413662920]\) | \(145993461081641846055362845817281/159280443487959447203639880\) | \(159280443487959447203639880\) | \([2]\) | \(55738368\) | \(3.9431\) | |
| 110670.co2 | 110670cm5 | \([1, 0, 0, -1096704020, 13979106944400]\) | \(145878744972947589224644168399681/189905470272000\) | \(189905470272000\) | \([6]\) | \(18579456\) | \(3.3937\) | |
| 110670.co3 | 110670cm6 | \([1, 0, 0, -85896020, 99386994000]\) | \(70087865855578132494813007681/36292294255990979366798400\) | \(36292294255990979366798400\) | \([2, 2]\) | \(27869184\) | \(3.5965\) | |
| 110670.co4 | 110670cm2 | \([1, 0, 0, -68544020, 218419136400]\) | \(35614957229424792235812559681/40635423166464000000\) | \(40635423166464000000\) | \([2, 6]\) | \(9289728\) | \(3.0472\) | |
| 110670.co5 | 110670cm4 | \([1, 0, 0, -67988500, 222133676432]\) | \(-34756024238867649947902344001/1203983372017125000000000\) | \(-1203983372017125000000000\) | \([6]\) | \(18579456\) | \(3.3937\) | |
| 110670.co6 | 110670cm3 | \([1, 0, 0, -48248340, -127876991088]\) | \(12421408380063888787371236161/124468522079571196907520\) | \(124468522079571196907520\) | \([2]\) | \(13934592\) | \(3.2499\) | |
| 110670.co7 | 110670cm1 | \([1, 0, 0, -4318740, 3354363792]\) | \(8908301264524959247997761/293465056522272768000\) | \(293465056522272768000\) | \([6]\) | \(4644864\) | \(2.7006\) | \(\Gamma_0(N)\)-optimal |
| 110670.co8 | 110670cm7 | \([1, 0, 0, 322836500, 772405961432]\) | \(3721102241752966083433176455999/2411133634213450079343165000\) | \(-2411133634213450079343165000\) | \([2]\) | \(55738368\) | \(3.9431\) |
Rank
sage: E.rank()
The elliptic curves in class 110670.co have rank \(0\).
Complex multiplication
The elliptic curves in class 110670.co do not have complex multiplication.Modular form 110670.2.a.co
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 6 & 12 & 4 & 12 & 4 \\ 3 & 1 & 6 & 2 & 4 & 12 & 4 & 12 \\ 2 & 6 & 1 & 3 & 6 & 2 & 6 & 2 \\ 6 & 2 & 3 & 1 & 2 & 6 & 2 & 6 \\ 12 & 4 & 6 & 2 & 1 & 12 & 4 & 3 \\ 4 & 12 & 2 & 6 & 12 & 1 & 3 & 4 \\ 12 & 4 & 6 & 2 & 4 & 3 & 1 & 12 \\ 4 & 12 & 2 & 6 & 3 & 4 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
