Show commands:
SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 177450dr
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177450.hb7 | 177450dr1 | \([1, 1, 1, -108688213, 433752569531]\) | \(1882742462388824401/11650189824000\) | \(878644392096744000000000\) | \([4]\) | \(37158912\) | \(3.4325\) | \(\Gamma_0(N)\)-optimal |
| 177450.hb6 | 177450dr2 | \([1, 1, 1, -174936213, -157577078469]\) | \(7850236389974007121/4400862921000000\) | \(331908199294517015625000000\) | \([2, 2]\) | \(74317824\) | \(3.7791\) | |
| 177450.hb5 | 177450dr3 | \([1, 1, 1, -671458213, -6405484770469]\) | \(443915739051786565201/21894701746029840\) | \(1651274116250821030946250000\) | \([4]\) | \(111476736\) | \(3.9818\) | |
| 177450.hb8 | 177450dr4 | \([1, 1, 1, 687470787, -1249384340469]\) | \(476437916651992691759/284661685546875000\) | \(-21468868527387908935546875000\) | \([2]\) | \(148635648\) | \(4.1257\) | |
| 177450.hb4 | 177450dr5 | \([1, 1, 1, -2097311213, -36909542328469]\) | \(13527956825588849127121/25701087819771000\) | \(1938347531222828761546875000\) | \([2]\) | \(148635648\) | \(4.1257\) | |
| 177450.hb2 | 177450dr6 | \([1, 1, 1, -10612798713, -420820204853469]\) | \(1752803993935029634719121/4599740941532100\) | \(346907358972743969826562500\) | \([2, 2]\) | \(222953472\) | \(4.3284\) | |
| 177450.hb3 | 177450dr7 | \([1, 1, 1, -10482372963, -431667192777969]\) | \(-1688971789881664420008241/89901485966373558750\) | \(-6780270337122899855258261718750\) | \([2]\) | \(445906944\) | \(4.6750\) | |
| 177450.hb1 | 177450dr8 | \([1, 1, 1, -169804672463, -26932316475430969]\) | \(7179471593960193209684686321/49441793310\) | \(3728845201950746718750\) | \([2]\) | \(445906944\) | \(4.6750\) |
Rank
sage: E.rank()
The elliptic curves in class 177450dr have rank \(1\).
Complex multiplication
The elliptic curves in class 177450dr do not have complex multiplication.Modular form 177450.2.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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
