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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 78078.bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
78078.bf1 | 78078bf4 | \([1, 0, 1, -2388481, -1420891834]\) | \(312196988566716625/25367712678\) | \(122445103863584502\) | \([2]\) | \(1327104\) | \(2.3240\) | |
78078.bf2 | 78078bf3 | \([1, 0, 1, -139091, -25370278]\) | \(-61653281712625/21875235228\) | \(-105587582275627452\) | \([2]\) | \(663552\) | \(1.9775\) | |
78078.bf3 | 78078bf2 | \([1, 0, 1, -61351, 2928434]\) | \(5290763640625/2291573592\) | \(11060988038027928\) | \([2]\) | \(442368\) | \(1.7747\) | |
78078.bf4 | 78078bf1 | \([1, 0, 1, 13009, 340706]\) | \(50447927375/39517632\) | \(-190744061796288\) | \([2]\) | \(221184\) | \(1.4282\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 78078.bf have rank \(1\).
Complex multiplication
The elliptic curves in class 78078.bf do not have complex multiplication.Modular form 78078.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.