Show commands:
SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 54450.cc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54450.cc1 | 54450bp6 | \([1, -1, 0, -4657789692, -122352764098284]\) | \(553808571467029327441/12529687500\) | \(252838907309838867187500\) | \([2]\) | \(35389440\) | \(4.0147\) | |
| 54450.cc2 | 54450bp4 | \([1, -1, 0, -321936192, 2218535595216]\) | \(182864522286982801/463015182960\) | \(9343269968023665753750000\) | \([2]\) | \(17694720\) | \(3.6682\) | |
| 54450.cc3 | 54450bp3 | \([1, -1, 0, -291444192, -1907123480784]\) | \(135670761487282321/643043610000\) | \(12976097265392626406250000\) | \([2, 2]\) | \(17694720\) | \(3.6682\) | |
| 54450.cc4 | 54450bp5 | \([1, -1, 0, -141706692, -3864342343284]\) | \(-15595206456730321/310672490129100\) | \(-6269118278925020699967187500\) | \([2]\) | \(35389440\) | \(4.0147\) | |
| 54450.cc5 | 54450bp2 | \([1, -1, 0, -27906192, 5371785216]\) | \(119102750067601/68309049600\) | \(1378421086738004100000000\) | \([2, 2]\) | \(8847360\) | \(3.3216\) | |
| 54450.cc6 | 54450bp1 | \([1, -1, 0, 6941808, 667305216]\) | \(1833318007919/1070530560\) | \(-21602436376181760000000\) | \([2]\) | \(4423680\) | \(2.9750\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 54450.cc have rank \(1\).
Complex multiplication
The elliptic curves in class 54450.cc do not have complex multiplication.Modular form 54450.2.a.cc
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
