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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 195195ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 195195.bl4 | 195195ca1 | \([1, 1, 0, -44788, 3629683]\) | \(2058561081361/12705\) | \(61324608345\) | \([2]\) | \(589824\) | \(1.2566\) | \(\Gamma_0(N)\)-optimal |
| 195195.bl3 | 195195ca2 | \([1, 1, 0, -45633, 3484512]\) | \(2177286259681/161417025\) | \(779129149023225\) | \([2, 2]\) | \(1179648\) | \(1.6032\) | |
| 195195.bl5 | 195195ca3 | \([1, 1, 0, 43092, 15497877]\) | \(1833318007919/22507682505\) | \(-108640284484276545\) | \([2]\) | \(2359296\) | \(1.9498\) | |
| 195195.bl2 | 195195ca4 | \([1, 1, 0, -147878, -17802897]\) | \(74093292126001/14707625625\) | \(70990899735380625\) | \([2, 2]\) | \(2359296\) | \(1.9498\) | |
| 195195.bl6 | 195195ca5 | \([1, 1, 0, 307577, -105341348]\) | \(666688497209279/1381398046875\) | \(-6667744525238671875\) | \([2]\) | \(4718592\) | \(2.2963\) | |
| 195195.bl1 | 195195ca6 | \([1, 1, 0, -2239253, -1290613722]\) | \(257260669489908001/14267882475\) | \(68868343541272275\) | \([2]\) | \(4718592\) | \(2.2963\) |
Rank
sage: E.rank()
The elliptic curves in class 195195ca have rank \(0\).
Complex multiplication
The elliptic curves in class 195195ca do not have complex multiplication.Modular form 195195.2.a.ca
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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
