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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 85800f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85800.d4 | 85800f1 | \([0, -1, 0, -1383, -2099988]\) | \(-1171019776/7623061875\) | \(-1905765468750000\) | \([2]\) | \(442368\) | \(1.6109\) | \(\Gamma_0(N)\)-optimal |
| 85800.d3 | 85800f2 | \([0, -1, 0, -254508, -48674988]\) | \(455795194086736/6998159025\) | \(27992636100000000\) | \([2, 2]\) | \(884736\) | \(1.9575\) | |
| 85800.d2 | 85800f3 | \([0, -1, 0, -502008, 61710012]\) | \(874453074310084/403786706895\) | \(6460587310320000000\) | \([4]\) | \(1769472\) | \(2.3040\) | |
| 85800.d1 | 85800f4 | \([0, -1, 0, -4057008, -3143909988]\) | \(461552841274085284/111344805\) | \(1781516880000000\) | \([2]\) | \(1769472\) | \(2.3040\) |
Rank
sage: E.rank()
The elliptic curves in class 85800f have rank \(0\).
Complex multiplication
The elliptic curves in class 85800f do not have complex multiplication.Modular form 85800.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
