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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 22050k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22050.c3 | 22050k1 | \([1, -1, 0, -9417, -398259]\) | \(-1860867/320\) | \(-15882615000000\) | \([2]\) | \(69120\) | \(1.2593\) | \(\Gamma_0(N)\)-optimal |
22050.c2 | 22050k2 | \([1, -1, 0, -156417, -23771259]\) | \(8527173507/200\) | \(9926634375000\) | \([2]\) | \(138240\) | \(1.6059\) | |
22050.c4 | 22050k3 | \([1, -1, 0, 64083, 1684241]\) | \(804357/500\) | \(-18091291148437500\) | \([2]\) | \(207360\) | \(1.8086\) | |
22050.c1 | 22050k4 | \([1, -1, 0, -266667, 13921991]\) | \(57960603/31250\) | \(1130705696777343750\) | \([2]\) | \(414720\) | \(2.1552\) |
Rank
sage: E.rank()
The elliptic curves in class 22050k have rank \(0\).
Complex multiplication
The elliptic curves in class 22050k do not have complex multiplication.Modular form 22050.2.a.k
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.