Show commands:
SageMath
E = EllipticCurve("ed1")
E.isogeny_class()
Elliptic curves in class 54720ed
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54720.i3 | 54720ed1 | \([0, 0, 0, -14901708, 22344518032]\) | \(-1914980734749238129/20440940544000\) | \(-3906324570197458944000\) | \([2]\) | \(4423680\) | \(2.9595\) | \(\Gamma_0(N)\)-optimal |
54720.i2 | 54720ed2 | \([0, 0, 0, -239034828, 1422459292048]\) | \(7903870428425797297009/886464000000\) | \(169405908516864000000\) | \([2]\) | \(8847360\) | \(3.3061\) | |
54720.i4 | 54720ed3 | \([0, 0, 0, 49241652, 116313213328]\) | \(69096190760262356111/70568821500000000\) | \(-13485911801462784000000000\) | \([2]\) | \(13271040\) | \(3.5088\) | |
54720.i1 | 54720ed4 | \([0, 0, 0, -266821068, 1071201902992]\) | \(10993009831928446009969/3767761230468750000\) | \(720030384000000000000000000\) | \([2]\) | \(26542080\) | \(3.8554\) |
Rank
sage: E.rank()
The elliptic curves in class 54720ed have rank \(0\).
Complex multiplication
The elliptic curves in class 54720ed do not have complex multiplication.Modular form 54720.2.a.ed
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.