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SageMath
E = EllipticCurve("jj1")
E.isogeny_class()
Elliptic curves in class 176400jj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
176400.d4 | 176400jj1 | \([0, 0, 0, 113925, 4030250]\) | \(804357/500\) | \(-101648736000000000\) | \([2]\) | \(1658880\) | \(1.9525\) | \(\Gamma_0(N)\)-optimal |
176400.d3 | 176400jj2 | \([0, 0, 0, -474075, 32842250]\) | \(57960603/31250\) | \(6353046000000000000\) | \([2]\) | \(3317760\) | \(2.2990\) | |
176400.d2 | 176400jj3 | \([0, 0, 0, -1356075, -692259750]\) | \(-1860867/320\) | \(-47425234268160000000\) | \([2]\) | \(4976640\) | \(2.5018\) | |
176400.d1 | 176400jj4 | \([0, 0, 0, -22524075, -41144307750]\) | \(8527173507/200\) | \(29640771417600000000\) | \([2]\) | \(9953280\) | \(2.8483\) |
Rank
sage: E.rank()
The elliptic curves in class 176400jj have rank \(1\).
Complex multiplication
The elliptic curves in class 176400jj do not have complex multiplication.Modular form 176400.2.a.jj
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.