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SageMath
E = EllipticCurve("mk1")
E.isogeny_class()
Elliptic curves in class 286650.mk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 286650.mk1 | 286650mk2 | \([1, -1, 1, -37077305, -86902395303]\) | \(-168256703745625/30371328\) | \(-1017512106319800000000\) | \([]\) | \(22394880\) | \(3.0347\) | |
| 286650.mk2 | 286650mk1 | \([1, -1, 1, 132070, -398040303]\) | \(7604375/2047032\) | \(-68580466485496875000\) | \([]\) | \(7464960\) | \(2.4854\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 286650.mk have rank \(1\).
Complex multiplication
The elliptic curves in class 286650.mk do not have complex multiplication.Modular form 286650.2.a.mk
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
