Show commands:
SageMath
E = EllipticCurve("lr1")
E.isogeny_class()
Elliptic curves in class 435600lr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
435600.lr6 | 435600lr1 | \([0, 0, 0, 111068925, -42818602750]\) | \(1833318007919/1070530560\) | \(-88483579396840488960000000\) | \([2]\) | \(106168320\) | \(3.6682\) | \(\Gamma_0(N)\)-optimal* |
435600.lr5 | 435600lr2 | \([0, 0, 0, -446499075, -343347754750]\) | \(119102750067601/68309049600\) | \(5646012771278864793600000000\) | \([2, 2]\) | \(212336640\) | \(4.0147\) | \(\Gamma_0(N)\)-optimal* |
435600.lr3 | 435600lr3 | \([0, 0, 0, -4663107075, 122060565877250]\) | \(135670761487282321/643043610000\) | \(53150094399048197760000000000\) | \([2, 2]\) | \(424673280\) | \(4.3613\) | \(\Gamma_0(N)\)-optimal* |
435600.lr2 | 435600lr4 | \([0, 0, 0, -5150979075, -141981127114750]\) | \(182864522286982801/463015182960\) | \(38270033789024934927360000000\) | \([2]\) | \(424673280\) | \(4.3613\) | |
435600.lr1 | 435600lr5 | \([0, 0, 0, -74524635075, 7830651426925250]\) | \(553808571467029327441/12529687500\) | \(1035628164341100000000000000\) | \([2]\) | \(849346560\) | \(4.7079\) | \(\Gamma_0(N)\)-optimal* |
435600.lr4 | 435600lr6 | \([0, 0, 0, -2267307075, 247320177277250]\) | \(-15595206456730321/310672490129100\) | \(-25678308470476884787065600000000\) | \([2]\) | \(849346560\) | \(4.7079\) |
Rank
sage: E.rank()
The elliptic curves in class 435600lr have rank \(0\).
Complex multiplication
The elliptic curves in class 435600lr do not have complex multiplication.Modular form 435600.2.a.lr
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.