Show commands:
SageMath
E = EllipticCurve("dc1")
E.isogeny_class()
Elliptic curves in class 34650dc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34650.dm3 | 34650dc1 | \([1, -1, 1, -12605, -10596603]\) | \(-19443408769/4249907200\) | \(-48409099200000000\) | \([2]\) | \(331776\) | \(1.8808\) | \(\Gamma_0(N)\)-optimal |
| 34650.dm2 | 34650dc2 | \([1, -1, 1, -804605, -275124603]\) | \(5057359576472449/51765560000\) | \(589642081875000000\) | \([2]\) | \(663552\) | \(2.2274\) | |
| 34650.dm4 | 34650dc3 | \([1, -1, 1, 113395, 285503397]\) | \(14156681599871/3100231750000\) | \(-35313577277343750000\) | \([2]\) | \(995328\) | \(2.4301\) | |
| 34650.dm1 | 34650dc4 | \([1, -1, 1, -5876105, 5328662397]\) | \(1969902499564819009/63690429687500\) | \(725473800659179687500\) | \([2]\) | \(1990656\) | \(2.7767\) |
Rank
sage: E.rank()
The elliptic curves in class 34650dc have rank \(0\).
Complex multiplication
The elliptic curves in class 34650dc do not have complex multiplication.Modular form 34650.2.a.dc
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.
