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SageMath
E = EllipticCurve("hd1")
E.isogeny_class()
Elliptic curves in class 47040.hd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.hd1 | 47040dk4 | \([0, 1, 0, -133345, -18777025]\) | \(68017239368/39375\) | \(151795445760000\) | \([2]\) | \(196608\) | \(1.6667\) | |
| 47040.hd2 | 47040dk3 | \([0, 1, 0, -78465, 8311743]\) | \(13858588808/229635\) | \(885271039672320\) | \([4]\) | \(196608\) | \(1.6667\) | |
| 47040.hd3 | 47040dk2 | \([0, 1, 0, -9865, -180937]\) | \(220348864/99225\) | \(47815565414400\) | \([2, 2]\) | \(98304\) | \(1.3201\) | |
| 47040.hd4 | 47040dk1 | \([0, 1, 0, 2140, -20070]\) | \(143877824/108045\) | \(-813528717120\) | \([2]\) | \(49152\) | \(0.97357\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 47040.hd have rank \(0\).
Complex multiplication
The elliptic curves in class 47040.hd do not have complex multiplication.Modular form 47040.2.a.hd
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.
