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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 51870e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51870.b4 | 51870e1 | \([1, 1, 0, 4153657, -4428885387]\) | \(7925295068364386256557831/13066712621053378560000\) | \(-13066712621053378560000\) | \([2]\) | \(4866048\) | \(2.9288\) | \(\Gamma_0(N)\)-optimal |
| 51870.b3 | 51870e2 | \([1, 1, 0, -28614343, -45474082187]\) | \(2591045694338778334837074169/606586705207219460505600\) | \(606586705207219460505600\) | \([2, 2]\) | \(9732096\) | \(3.2753\) | |
| 51870.b2 | 51870e3 | \([1, 1, 0, -153375943, 692540686453]\) | \(399020725834811692396780472569/23683887027953031545003520\) | \(23683887027953031545003520\) | \([2]\) | \(19464192\) | \(3.6219\) | |
| 51870.b1 | 51870e4 | \([1, 1, 0, -428140743, -3409726086027]\) | \(8679273798268409608241895147769/721206017935072932487680\) | \(721206017935072932487680\) | \([2]\) | \(19464192\) | \(3.6219\) |
Rank
sage: E.rank()
The elliptic curves in class 51870e have rank \(1\).
Complex multiplication
The elliptic curves in class 51870e do not have complex multiplication.Modular form 51870.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
