Show commands:
SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 172480.dr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
172480.dr1 | 172480bs4 | \([0, 0, 0, -65571212, -204370359536]\) | \(1010962818911303721/57392720\) | \(1770047698443960320\) | \([2]\) | \(9437184\) | \(2.9681\) | |
172480.dr2 | 172480bs3 | \([0, 0, 0, -6865292, 1632828176]\) | \(1160306142246441/634128110000\) | \(19557132012982108160000\) | \([2]\) | \(9437184\) | \(2.9681\) | |
172480.dr3 | 172480bs2 | \([0, 0, 0, -4105612, -3181157616]\) | \(248158561089321/1859334400\) | \(57343694032234086400\) | \([2, 2]\) | \(4718592\) | \(2.6215\) | |
172480.dr4 | 172480bs1 | \([0, 0, 0, -91532, -112794864]\) | \(-2749884201/176619520\) | \(-5447118987848581120\) | \([2]\) | \(2359296\) | \(2.2749\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 172480.dr have rank \(0\).
Complex multiplication
The elliptic curves in class 172480.dr do not have complex multiplication.Modular form 172480.2.a.dr
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.