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SageMath
E = EllipticCurve("du1")
E.isogeny_class()
Elliptic curves in class 85680du
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 85680.c3 | 85680du1 | \([0, 0, 0, -146163, -21033038]\) | \(115650783909361/2924544000\) | \(8732641591296000\) | \([2]\) | \(589824\) | \(1.8416\) | \(\Gamma_0(N)\)-optimal |
| 85680.c2 | 85680du2 | \([0, 0, 0, -330483, 42704818]\) | \(1336852858103281/509796000000\) | \(1522242699264000000\) | \([2, 2]\) | \(1179648\) | \(2.1882\) | |
| 85680.c4 | 85680du3 | \([0, 0, 0, 1040397, 305639602]\) | \(41709358422320399/37652343750000\) | \(-112429296000000000000\) | \([2]\) | \(2359296\) | \(2.5348\) | |
| 85680.c1 | 85680du4 | \([0, 0, 0, -4650483, 3858992818]\) | \(3725035528036823281/1203203526000\) | \(3592746477379584000\) | \([2]\) | \(2359296\) | \(2.5348\) |
Rank
sage: E.rank()
The elliptic curves in class 85680du have rank \(1\).
Complex multiplication
The elliptic curves in class 85680du do not have complex multiplication.Modular form 85680.2.a.du
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
