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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 201810.ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 201810.ca1 | 201810bb7 | \([1, 1, 1, -1845888820, 30524283602207]\) | \(783736670177727068275201/360150\) | \(319634450712150\) | \([2]\) | \(62914560\) | \(3.5114\) | |
| 201810.ca2 | 201810bb5 | \([1, 1, 1, -115368070, 476905715807]\) | \(191342053882402567201/129708022500\) | \(115116347423980822500\) | \([2, 2]\) | \(31457280\) | \(3.1648\) | |
| 201810.ca3 | 201810bb8 | \([1, 1, 1, -114647320, 483159519407]\) | \(-187778242790732059201/4984939585440150\) | \(-4424152231640747130192150\) | \([2]\) | \(62914560\) | \(3.5114\) | |
| 201810.ca4 | 201810bb4 | \([1, 1, 1, -14482290, -21213288945]\) | \(378499465220294881/120530818800\) | \(106971545358944002800\) | \([2]\) | \(15728640\) | \(2.8183\) | |
| 201810.ca5 | 201810bb3 | \([1, 1, 1, -7255570, 7351505807]\) | \(47595748626367201/1215506250000\) | \(1078766271153506250000\) | \([2, 2]\) | \(15728640\) | \(2.8183\) | |
| 201810.ca6 | 201810bb2 | \([1, 1, 1, -1028290, -235812145]\) | \(135487869158881/51438240000\) | \(45651627344161440000\) | \([2, 2]\) | \(7864320\) | \(2.4717\) | |
| 201810.ca7 | 201810bb1 | \([1, 1, 1, 201790, -26206513]\) | \(1023887723039/928972800\) | \(-824466779548876800\) | \([2]\) | \(3932160\) | \(2.1251\) | \(\Gamma_0(N)\)-optimal |
| 201810.ca8 | 201810bb6 | \([1, 1, 1, 1220450, 23510190335]\) | \(226523624554079/269165039062500\) | \(-238884962964477539062500\) | \([2]\) | \(31457280\) | \(3.1648\) |
Rank
sage: E.rank()
The elliptic curves in class 201810.ca have rank \(0\).
Complex multiplication
The elliptic curves in class 201810.ca do not have complex multiplication.Modular form 201810.2.a.ca
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
