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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 42350.ba
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42350.ba1 | 42350u4 | \([1, -1, 0, -63250292, 193632103616]\) | \(1010962818911303721/57392720\) | \(1588667256811250000\) | \([2]\) | \(2949120\) | \(2.9591\) | |
42350.ba2 | 42350u3 | \([1, -1, 0, -6622292, -1545252384]\) | \(1160306142246441/634128110000\) | \(17553072323120468750000\) | \([2]\) | \(2949120\) | \(2.9591\) | |
42350.ba3 | 42350u2 | \([1, -1, 0, -3960292, 3014753616]\) | \(248158561089321/1859334400\) | \(51467567328100000000\) | \([2, 2]\) | \(1474560\) | \(2.6125\) | |
42350.ba4 | 42350u1 | \([1, -1, 0, -88292, 106881616]\) | \(-2749884201/176619520\) | \(-4888941460480000000\) | \([2]\) | \(737280\) | \(2.2659\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 42350.ba have rank \(1\).
Complex multiplication
The elliptic curves in class 42350.ba do not have complex multiplication.Modular form 42350.2.a.ba
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.