Show commands:
SageMath
E = EllipticCurve("du1")
E.isogeny_class()
Elliptic curves in class 172480du
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 172480.gf3 | 172480du1 | \([0, -1, 0, -175681, 551610081]\) | \(-19443408769/4249907200\) | \(-131071300645106483200\) | \([2]\) | \(5308416\) | \(2.5395\) | \(\Gamma_0(N)\)-optimal |
| 172480.gf2 | 172480du2 | \([0, -1, 0, -11214401, 14330140385]\) | \(5057359576472449/51765560000\) | \(1596500572488335360000\) | \([2]\) | \(10616832\) | \(2.8860\) | |
| 172480.gf4 | 172480du3 | \([0, -1, 0, 1580479, -14857991455]\) | \(14156681599871/3100231750000\) | \(-95614183710588928000000\) | \([2]\) | \(15925248\) | \(3.0888\) | |
| 172480.gf1 | 172480du4 | \([0, -1, 0, -81899841, -277169852959]\) | \(1969902499564819009/63690429687500\) | \(1964275233536000000000000\) | \([2]\) | \(31850496\) | \(3.4353\) |
Rank
sage: E.rank()
The elliptic curves in class 172480du have rank \(1\).
Complex multiplication
The elliptic curves in class 172480du do not have complex multiplication.Modular form 172480.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
