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SageMath
E = EllipticCurve("fh1")
E.isogeny_class()
Elliptic curves in class 47040.fh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
47040.fh1 | 47040cm4 | \([0, 1, 0, -63507201, 194775968799]\) | \(7347751505995469192/72930375\) | \(281155524636672000\) | \([2]\) | \(2949120\) | \(2.9244\) | |
47040.fh2 | 47040cm3 | \([0, 1, 0, -5687201, 157376799]\) | \(5276930158229192/3050936350875\) | \(11761733164862435328000\) | \([2]\) | \(2949120\) | \(2.9244\) | |
47040.fh3 | 47040cm2 | \([0, 1, 0, -3972201, 3037547799]\) | \(14383655824793536/45209390625\) | \(21785966991936000000\) | \([2, 2]\) | \(1474560\) | \(2.5778\) | |
47040.fh4 | 47040cm1 | \([0, 1, 0, -144076, 87594674]\) | \(-43927191786304/415283203125\) | \(-3126889828125000000\) | \([2]\) | \(737280\) | \(2.2313\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 47040.fh have rank \(1\).
Complex multiplication
The elliptic curves in class 47040.fh do not have complex multiplication.Modular form 47040.2.a.fh
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.