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SageMath
E = EllipticCurve("be1")
E.isogeny_class()
Elliptic curves in class 68208.be
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68208.be1 | 68208bw4 | \([0, -1, 0, -160425232, 782144188288]\) | \(947531277805646290177/38367\) | \(18488685293568\) | \([2]\) | \(4718592\) | \(2.9568\) | |
| 68208.be2 | 68208bw6 | \([0, -1, 0, -33295712, -60103232832]\) | \(8471112631466271697/1662662681263647\) | \(801221024923593956978688\) | \([2]\) | \(9437184\) | \(3.3033\) | |
| 68208.be3 | 68208bw3 | \([0, -1, 0, -10218672, 11730977280]\) | \(244883173420511137/18418027974129\) | \(8875469099533527945216\) | \([2, 2]\) | \(4718592\) | \(2.9568\) | |
| 68208.be4 | 68208bw2 | \([0, -1, 0, -10026592, 12223470400]\) | \(231331938231569617/1472026689\) | \(709355388658323456\) | \([2, 2]\) | \(2359296\) | \(2.6102\) | |
| 68208.be5 | 68208bw1 | \([0, -1, 0, -614672, 198801408]\) | \(-53297461115137/4513839183\) | \(-2175175336102981632\) | \([2]\) | \(1179648\) | \(2.2636\) | \(\Gamma_0(N)\)-optimal |
| 68208.be6 | 68208bw5 | \([0, -1, 0, 9785088, 52042554432]\) | \(215015459663151503/2552757445339983\) | \(-1230149061373147791224832\) | \([4]\) | \(9437184\) | \(3.3033\) |
Rank
sage: E.rank()
The elliptic curves in class 68208.be have rank \(0\).
Complex multiplication
The elliptic curves in class 68208.be do not have complex multiplication.Modular form 68208.2.a.be
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.
