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SageMath
E = EllipticCurve("kh1")
E.isogeny_class()
Elliptic curves in class 277200.kh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 277200.kh1 | 277200kh4 | \([0, 0, 0, -94017675, -340940375750]\) | \(1969902499564819009/63690429687500\) | \(2971540687500000000000000\) | \([2]\) | \(47775744\) | \(3.4698\) | |
| 277200.kh2 | 277200kh2 | \([0, 0, 0, -12873675, 17620848250]\) | \(5057359576472449/51765560000\) | \(2415173967360000000000\) | \([2]\) | \(15925248\) | \(2.9205\) | |
| 277200.kh3 | 277200kh1 | \([0, 0, 0, -201675, 678384250]\) | \(-19443408769/4249907200\) | \(-198283670323200000000\) | \([2]\) | \(7962624\) | \(2.5740\) | \(\Gamma_0(N)\)-optimal |
| 277200.kh4 | 277200kh3 | \([0, 0, 0, 1814325, -18274031750]\) | \(14156681599871/3100231750000\) | \(-144644412528000000000000\) | \([2]\) | \(23887872\) | \(3.1233\) |
Rank
sage: E.rank()
The elliptic curves in class 277200.kh have rank \(0\).
Complex multiplication
The elliptic curves in class 277200.kh do not have complex multiplication.Modular form 277200.2.a.kh
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
