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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 22386.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22386.m1 | 22386i4 | \([1, 0, 1, -90975, -10569134]\) | \(83268941223547539433/3317067936\) | \(3317067936\) | \([2]\) | \(69120\) | \(1.3155\) | |
22386.m2 | 22386i2 | \([1, 0, 1, -5695, -164974]\) | \(20421858870283753/128290046976\) | \(128290046976\) | \([2, 2]\) | \(34560\) | \(0.96897\) | |
22386.m3 | 22386i3 | \([1, 0, 1, -2335, -357166]\) | \(-1407074115849193/54234808266912\) | \(-54234808266912\) | \([2]\) | \(69120\) | \(1.3155\) | |
22386.m4 | 22386i1 | \([1, 0, 1, -575, 914]\) | \(20972058349033/11736711168\) | \(11736711168\) | \([2]\) | \(17280\) | \(0.62239\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 22386.m have rank \(1\).
Complex multiplication
The elliptic curves in class 22386.m do not have complex multiplication.Modular form 22386.2.a.m
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.