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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 33810.cc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33810.cc1 | 33810bz6 | \([1, 1, 1, -4149811, 3252065123]\) | \(67176973097223766561/91487391870\) | \(10763400166113630\) | \([2]\) | \(786432\) | \(2.3504\) | |
| 33810.cc2 | 33810bz4 | \([1, 1, 1, -261661, 49784783]\) | \(16840406336564161/604708416900\) | \(71143340539868100\) | \([2, 2]\) | \(393216\) | \(2.0038\) | |
| 33810.cc3 | 33810bz2 | \([1, 1, 1, -41161, -2165017]\) | \(65553197996161/20996010000\) | \(2470159580490000\) | \([2, 2]\) | \(196608\) | \(1.6572\) | |
| 33810.cc4 | 33810bz1 | \([1, 1, 1, -37241, -2781241]\) | \(48551226272641/9273600\) | \(1091029766400\) | \([2]\) | \(98304\) | \(1.3107\) | \(\Gamma_0(N)\)-optimal |
| 33810.cc5 | 33810bz5 | \([1, 1, 1, 98489, 176701643]\) | \(898045580910239/115117148363070\) | \(-13543417387766822430\) | \([2]\) | \(786432\) | \(2.3504\) | |
| 33810.cc6 | 33810bz3 | \([1, 1, 1, 116619, -14598081]\) | \(1490881681033919/1650501562500\) | \(-194179858326562500\) | \([2]\) | \(393216\) | \(2.0038\) |
Rank
sage: E.rank()
The elliptic curves in class 33810.cc have rank \(1\).
Complex multiplication
The elliptic curves in class 33810.cc do not have complex multiplication.Modular form 33810.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
