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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 450d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
450.c3 | 450d1 | \([1, -1, 0, -27, -59]\) | \(-121945/32\) | \(-583200\) | \([]\) | \(60\) | \(-0.17679\) | \(\Gamma_0(N)\)-optimal |
450.c4 | 450d2 | \([1, -1, 0, 198, 436]\) | \(46969655/32768\) | \(-597196800\) | \([]\) | \(180\) | \(0.37251\) | |
450.c2 | 450d3 | \([1, -1, 0, -117, 5791]\) | \(-25/2\) | \(-14238281250\) | \([]\) | \(300\) | \(0.62793\) | |
450.c1 | 450d4 | \([1, -1, 0, -28242, 1833916]\) | \(-349938025/8\) | \(-56953125000\) | \([]\) | \(900\) | \(1.1772\) |
Rank
sage: E.rank()
The elliptic curves in class 450d have rank \(0\).
Complex multiplication
The elliptic curves in class 450d do not have complex multiplication.Modular form 450.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 5 & 15 \\ 3 & 1 & 15 & 5 \\ 5 & 15 & 1 & 3 \\ 15 & 5 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.