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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 75810.q
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 75810.q1 | 75810x8 | \([1, 1, 0, -693408807, 7027717411239]\) | \(783736670177727068275201/360150\) | \(16943574042150\) | \([2]\) | \(14155776\) | \(3.2666\) | |
| 75810.q2 | 75810x6 | \([1, 1, 0, -43338057, 109794503889]\) | \(191342053882402567201/129708022500\) | \(6102228191280322500\) | \([2, 2]\) | \(7077888\) | \(2.9201\) | |
| 75810.q3 | 75810x7 | \([1, 1, 0, -43067307, 111234406539]\) | \(-187778242790732059201/4984939585440150\) | \(-234520874528806629522150\) | \([2]\) | \(14155776\) | \(3.2666\) | |
| 75810.q4 | 75810x4 | \([1, 1, 0, -5440277, -4884968259]\) | \(378499465220294881/120530818800\) | \(5670478558097362800\) | \([2]\) | \(3538944\) | \(2.5735\) | |
| 75810.q5 | 75810x3 | \([1, 1, 0, -2725557, 1692151389]\) | \(47595748626367201/1215506250000\) | \(57184562392256250000\) | \([2, 2]\) | \(3538944\) | \(2.5735\) | |
| 75810.q6 | 75810x2 | \([1, 1, 0, -386277, -54355059]\) | \(135487869158881/51438240000\) | \(2419957317889440000\) | \([2, 2]\) | \(1769472\) | \(2.2269\) | |
| 75810.q7 | 75810x1 | \([1, 1, 0, 75803, -6021491]\) | \(1023887723039/928972800\) | \(-43704343801036800\) | \([2]\) | \(884736\) | \(1.8804\) | \(\Gamma_0(N)\)-optimal |
| 75810.q8 | 75810x5 | \([1, 1, 0, 458463, 5412997161]\) | \(226523624554079/269165039062500\) | \(-12663106397094726562500\) | \([2]\) | \(7077888\) | \(2.9201\) |
Rank
sage: E.rank()
The elliptic curves in class 75810.q have rank \(1\).
Complex multiplication
The elliptic curves in class 75810.q do not have complex multiplication.Modular form 75810.2.a.q
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
