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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1694.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1694.e1 | 1694f6 | \([1, 0, 0, -330393, 73068601]\) | \(2251439055699625/25088\) | \(44444922368\) | \([2]\) | \(8640\) | \(1.6120\) | |
| 1694.e2 | 1694f5 | \([1, 0, 0, -20633, 1142329]\) | \(-548347731625/1835008\) | \(-3250828607488\) | \([2]\) | \(4320\) | \(1.2655\) | |
| 1694.e3 | 1694f4 | \([1, 0, 0, -4298, 88540]\) | \(4956477625/941192\) | \(1667379040712\) | \([2]\) | \(2880\) | \(1.0627\) | |
| 1694.e4 | 1694f2 | \([1, 0, 0, -1273, -17577]\) | \(128787625/98\) | \(173612978\) | \([2]\) | \(960\) | \(0.51344\) | |
| 1694.e5 | 1694f1 | \([1, 0, 0, -63, -395]\) | \(-15625/28\) | \(-49603708\) | \([2]\) | \(480\) | \(0.16686\) | \(\Gamma_0(N)\)-optimal |
| 1694.e6 | 1694f3 | \([1, 0, 0, 542, 8196]\) | \(9938375/21952\) | \(-38889307072\) | \([2]\) | \(1440\) | \(0.71617\) |
Rank
sage: E.rank()
The elliptic curves in class 1694.e have rank \(1\).
Complex multiplication
The elliptic curves in class 1694.e do not have complex multiplication.Modular form 1694.2.a.e
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
