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SageMath
E = EllipticCurve("fm1")
E.isogeny_class()
Elliptic curves in class 111600fm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
111600.gi3 | 111600fm1 | \([0, 0, 0, -876675, 315909250]\) | \(1597099875769/186000\) | \(8678016000000000\) | \([2]\) | \(1327104\) | \(2.0846\) | \(\Gamma_0(N)\)-optimal |
111600.gi2 | 111600fm2 | \([0, 0, 0, -948675, 260973250]\) | \(2023804595449/540562500\) | \(25220484000000000000\) | \([2, 2]\) | \(2654208\) | \(2.4312\) | |
111600.gi4 | 111600fm3 | \([0, 0, 0, 2399325, 1690569250]\) | \(32740359775271/45410156250\) | \(-2118656250000000000000\) | \([2]\) | \(5308416\) | \(2.7778\) | |
111600.gi1 | 111600fm4 | \([0, 0, 0, -5448675, -4684526750]\) | \(383432500775449/18701300250\) | \(872527864464000000000\) | \([2]\) | \(5308416\) | \(2.7778\) |
Rank
sage: E.rank()
The elliptic curves in class 111600fm have rank \(0\).
Complex multiplication
The elliptic curves in class 111600fm do not have complex multiplication.Modular form 111600.2.a.fm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.