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SageMath
E = EllipticCurve("cv1")
E.isogeny_class()
Elliptic curves in class 196350cv
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 196350.dv7 | 196350cv1 | \([1, 1, 1, -23641338, -370830969]\) | \(93523304529581769096409/54118679989886265600\) | \(845604374841972900000000\) | \([2]\) | \(31850496\) | \(3.2807\) | \(\Gamma_0(N)\)-optimal |
| 196350.dv5 | 196350cv2 | \([1, 1, 1, -258939338, 1598714377031]\) | \(122884692280581205924284889/439106354595306090000\) | \(6861036790551657656250000\) | \([2, 2]\) | \(63700992\) | \(3.6272\) | |
| 196350.dv4 | 196350cv3 | \([1, 1, 1, -1321081713, -18482181177969]\) | \(16318969429297971769640983369/102045248126976000000\) | \(1594457001984000000000000\) | \([2]\) | \(95551488\) | \(3.8300\) | |
| 196350.dv2 | 196350cv4 | \([1, 1, 1, -4139469838, 102508029499031]\) | \(502039459750388822744052370969/6444603154532812500\) | \(100696924289575195312500\) | \([2]\) | \(127401984\) | \(3.9738\) | |
| 196350.dv6 | 196350cv5 | \([1, 1, 1, -143176838, 3035558527031]\) | \(-20774088968758822168212889/242753662862303369030100\) | \(-3793025982223490141095312500\) | \([2]\) | \(127401984\) | \(3.9738\) | |
| 196350.dv3 | 196350cv6 | \([1, 1, 1, -1346169713, -17743740985969]\) | \(17266453047612484705388895049/1288004819409000000000000\) | \(20125075303265625000000000000\) | \([2, 2]\) | \(191102976\) | \(4.1765\) | |
| 196350.dv1 | 196350cv7 | \([1, 1, 1, -4372577713, 90317183062031]\) | \(591720065532918583239955136329/116891407012939453125000000\) | \(1826428234577178955078125000000\) | \([2]\) | \(382205952\) | \(4.5231\) | |
| 196350.dv8 | 196350cv8 | \([1, 1, 1, 1278830287, -78543990985969]\) | \(14802750729576629005731104951/179133615680899546821000000\) | \(-2798962745014055419078125000000\) | \([2]\) | \(382205952\) | \(4.5231\) |
Rank
sage: E.rank()
The elliptic curves in class 196350cv have rank \(0\).
Complex multiplication
The elliptic curves in class 196350cv do not have complex multiplication.Modular form 196350.2.a.cv
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
