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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 23534e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 23534.o5 | 23534e1 | \([1, 1, 0, -875, 19817]\) | \(-15625/28\) | \(-133002918748\) | \([2]\) | \(23040\) | \(0.82470\) | \(\Gamma_0(N)\)-optimal |
| 23534.o4 | 23534e2 | \([1, 1, 0, -17685, 897299]\) | \(128787625/98\) | \(465510215618\) | \([2]\) | \(46080\) | \(1.1713\) | |
| 23534.o6 | 23534e3 | \([1, 1, 0, 7530, -418924]\) | \(9938375/21952\) | \(-104274288298432\) | \([2]\) | \(69120\) | \(1.3740\) | |
| 23534.o3 | 23534e4 | \([1, 1, 0, -59710, -4628148]\) | \(4956477625/941192\) | \(4470760110795272\) | \([2]\) | \(138240\) | \(1.7206\) | |
| 23534.o2 | 23534e5 | \([1, 1, 0, -286645, -59359827]\) | \(-548347731625/1835008\) | \(-8716479283068928\) | \([2]\) | \(207360\) | \(1.9233\) | |
| 23534.o1 | 23534e6 | \([1, 1, 0, -4590005, -3786930259]\) | \(2251439055699625/25088\) | \(119170615198208\) | \([2]\) | \(414720\) | \(2.2699\) |
Rank
sage: E.rank()
The elliptic curves in class 23534e have rank \(1\).
Complex multiplication
The elliptic curves in class 23534e do not have complex multiplication.Modular form 23534.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
