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SageMath
E = EllipticCurve("cc1")
E.isogeny_class()
Elliptic curves in class 76050.cc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 76050.cc1 | 76050h4 | \([1, -1, 0, -4855317, -4116590659]\) | \(8527173507/200\) | \(296894004834375000\) | \([2]\) | \(2488320\) | \(2.4647\) | |
| 76050.cc2 | 76050h3 | \([1, -1, 0, -292317, -69209659]\) | \(-1860867/320\) | \(-475030407735000000\) | \([2]\) | \(1244160\) | \(2.1181\) | |
| 76050.cc3 | 76050h2 | \([1, -1, 0, -102192, 3312466]\) | \(57960603/31250\) | \(63634688964843750\) | \([2]\) | \(829440\) | \(1.9154\) | |
| 76050.cc4 | 76050h1 | \([1, -1, 0, 24558, 397216]\) | \(804357/500\) | \(-1018155023437500\) | \([2]\) | \(414720\) | \(1.5688\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 76050.cc have rank \(1\).
Complex multiplication
The elliptic curves in class 76050.cc do not have complex multiplication.Modular form 76050.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
