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SageMath
E = EllipticCurve("du1")
E.isogeny_class()
Elliptic curves in class 185130du
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 185130.b4 | 185130du1 | \([1, -1, 0, 8145, -1884195]\) | \(46268279/1211760\) | \(-1564949226115440\) | \([2]\) | \(983040\) | \(1.5957\) | \(\Gamma_0(N)\)-optimal |
| 185130.b3 | 185130du2 | \([1, -1, 0, -187875, -29758239]\) | \(567869252041/31472100\) | \(40645209067164900\) | \([2, 2]\) | \(1966080\) | \(1.9423\) | |
| 185130.b2 | 185130du3 | \([1, -1, 0, -547245, 118374075]\) | \(14034143923561/3445241250\) | \(4449418719852521250\) | \([2]\) | \(3932160\) | \(2.2888\) | |
| 185130.b1 | 185130du4 | \([1, -1, 0, -2964825, -1964181609]\) | \(2231707882611241/7466910\) | \(9643275092405790\) | \([2]\) | \(3932160\) | \(2.2888\) |
Rank
sage: E.rank()
The elliptic curves in class 185130du have rank \(1\).
Complex multiplication
The elliptic curves in class 185130du do not have complex multiplication.Modular form 185130.2.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.
