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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 35490.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 35490.a1 | 35490b6 | \([1, 1, 0, -2839203, -1842560793]\) | \(524388516989299201/3150\) | \(15204448350\) | \([2]\) | \(589824\) | \(2.0192\) | |
| 35490.a2 | 35490b4 | \([1, 1, 0, -177453, -28844343]\) | \(128031684631201/9922500\) | \(47894012302500\) | \([2, 2]\) | \(294912\) | \(1.6727\) | |
| 35490.a3 | 35490b5 | \([1, 1, 0, -165623, -32840517]\) | \(-104094944089921/35880468750\) | \(-173188169486718750\) | \([2]\) | \(589824\) | \(2.0192\) | |
| 35490.a4 | 35490b3 | \([1, 1, 0, -62533, 5662753]\) | \(5602762882081/345888060\) | \(1669535601000540\) | \([2]\) | \(294912\) | \(1.6727\) | |
| 35490.a5 | 35490b2 | \([1, 1, 0, -11833, -390827]\) | \(37966934881/8643600\) | \(41721006272400\) | \([2, 2]\) | \(147456\) | \(1.3261\) | |
| 35490.a6 | 35490b1 | \([1, 1, 0, 1687, -36603]\) | \(109902239/188160\) | \(-908212381440\) | \([2]\) | \(73728\) | \(0.97951\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 35490.a have rank \(1\).
Complex multiplication
The elliptic curves in class 35490.a do not have complex multiplication.Modular form 35490.2.a.a
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.
