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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 40362.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
40362.k1 | 40362j2 | \([1, 1, 0, -635721, 4347809733]\) | \(-32015057794777/9184009519104\) | \(-8150842254543839821824\) | \([]\) | \(2903040\) | \(2.8836\) | |
40362.k2 | 40362j1 | \([1, 1, 0, 70614, -160726572]\) | \(43874924183/12604443264\) | \(-11186489793755654784\) | \([]\) | \(967680\) | \(2.3343\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 40362.k have rank \(1\).
Complex multiplication
The elliptic curves in class 40362.k do not have complex multiplication.Modular form 40362.2.a.k
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.