Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 2850a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2850.e4 | 2850a1 | \([1, 1, 0, 233375, -92172875]\) | \(89962967236397039/287450726400000\) | \(-4491417600000000000\) | \([2]\) | \(57600\) | \(2.2623\) | \(\Gamma_0(N)\)-optimal |
| 2850.e3 | 2850a2 | \([1, 1, 0, -2198625, -1081996875]\) | \(75224183150104868881/11219310000000000\) | \(175301718750000000000\) | \([2]\) | \(115200\) | \(2.6088\) | |
| 2850.e2 | 2850a3 | \([1, 1, 0, -82536625, -288649002875]\) | \(-3979640234041473454886161/1471455901872240\) | \(-22991498466753750000\) | \([2]\) | \(288000\) | \(3.0670\) | |
| 2850.e1 | 2850a4 | \([1, 1, 0, -1320586125, -18471882009375]\) | \(16300610738133468173382620881/2228489100\) | \(34820142187500\) | \([2]\) | \(576000\) | \(3.4135\) |
Rank
sage: E.rank()
The elliptic curves in class 2850a have rank \(1\).
Complex multiplication
The elliptic curves in class 2850a do not have complex multiplication.Modular form 2850.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
