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SageMath
E = EllipticCurve("cy1")
E.isogeny_class()
Elliptic curves in class 22800cy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22800.df3 | 22800cy1 | \([0, 1, 0, -10348408, -12934304812]\) | \(-1914980734749238129/20440940544000\) | \(-1308220194816000000000\) | \([2]\) | \(1658880\) | \(2.8683\) | \(\Gamma_0(N)\)-optimal |
22800.df2 | 22800cy2 | \([0, 1, 0, -165996408, -823237792812]\) | \(7903870428425797297009/886464000000\) | \(56733696000000000000\) | \([2]\) | \(3317760\) | \(3.2149\) | |
22800.df4 | 22800cy3 | \([0, 1, 0, 34195592, -67299488812]\) | \(69096190760262356111/70568821500000000\) | \(-4516404576000000000000000\) | \([2]\) | \(4976640\) | \(3.4177\) | |
22800.df1 | 22800cy4 | \([0, 1, 0, -185292408, -619970272812]\) | \(10993009831928446009969/3767761230468750000\) | \(241136718750000000000000000\) | \([2]\) | \(9953280\) | \(3.7642\) |
Rank
sage: E.rank()
The elliptic curves in class 22800cy have rank \(1\).
Complex multiplication
The elliptic curves in class 22800cy do not have complex multiplication.Modular form 22800.2.a.cy
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.