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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 122010.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
122010.d1 | 122010e2 | \([1, 1, 0, -928673, 344215677]\) | \(-752878092784212361/353021293800\) | \(-41532602194276200\) | \([]\) | \(2073600\) | \(2.1447\) | |
122010.d2 | 122010e1 | \([1, 1, 0, 8452, 1878402]\) | \(567457901639/13235906250\) | \(-1557191134406250\) | \([]\) | \(691200\) | \(1.5954\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 122010.d have rank \(1\).
Complex multiplication
The elliptic curves in class 122010.d do not have complex multiplication.Modular form 122010.2.a.d
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.