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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 272832.o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 272832.o1 | 272832o4 | \([0, -1, 0, -641700929, -6256511805375]\) | \(947531277805646290177/38367\) | \(1183275858788352\) | \([2]\) | \(37748736\) | \(3.3033\) | |
| 272832.o2 | 272832o5 | \([0, -1, 0, -133182849, 480959045505]\) | \(8471112631466271697/1662662681263647\) | \(51278145595110013246636032\) | \([2]\) | \(75497472\) | \(3.6499\) | |
| 272832.o3 | 272832o3 | \([0, -1, 0, -40874689, -93806943551]\) | \(244883173420511137/18418027974129\) | \(568030022370145788493824\) | \([2, 2]\) | \(37748736\) | \(3.3033\) | |
| 272832.o4 | 272832o2 | \([0, -1, 0, -40106369, -97747656831]\) | \(231331938231569617/1472026689\) | \(45398744874132701184\) | \([2, 2]\) | \(18874368\) | \(2.9568\) | |
| 272832.o5 | 272832o1 | \([0, -1, 0, -2458689, -1587952575]\) | \(-53297461115137/4513839183\) | \(-139211221510590824448\) | \([2]\) | \(9437184\) | \(2.6102\) | \(\Gamma_0(N)\)-optimal |
| 272832.o6 | 272832o6 | \([0, -1, 0, 39140351, -416379575807]\) | \(215015459663151503/2552757445339983\) | \(-78729539927881458638389248\) | \([2]\) | \(75497472\) | \(3.6499\) |
Rank
sage: E.rank()
The elliptic curves in class 272832.o have rank \(0\).
Complex multiplication
The elliptic curves in class 272832.o do not have complex multiplication.Modular form 272832.2.a.o
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
