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SageMath
E = EllipticCurve("fi1")
E.isogeny_class()
Elliptic curves in class 68400.fi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68400.fi1 | 68400ej4 | \([0, 0, 0, -1667631675, 16737529734250]\) | \(10993009831928446009969/3767761230468750000\) | \(175788667968750000000000000000\) | \([2]\) | \(79626240\) | \(4.3135\) | |
68400.fi2 | 68400ej2 | \([0, 0, 0, -1493967675, 22225926438250]\) | \(7903870428425797297009/886464000000\) | \(41358864384000000000000\) | \([2]\) | \(26542080\) | \(3.7642\) | |
68400.fi3 | 68400ej1 | \([0, 0, 0, -93135675, 349133094250]\) | \(-1914980734749238129/20440940544000\) | \(-953692522020864000000000\) | \([2]\) | \(13271040\) | \(3.4177\) | \(\Gamma_0(N)\)-optimal |
68400.fi4 | 68400ej3 | \([0, 0, 0, 307760325, 1817393958250]\) | \(69096190760262356111/70568821500000000\) | \(-3292458935904000000000000000\) | \([2]\) | \(39813120\) | \(3.9670\) |
Rank
sage: E.rank()
The elliptic curves in class 68400.fi have rank \(1\).
Complex multiplication
The elliptic curves in class 68400.fi do not have complex multiplication.Modular form 68400.2.a.fi
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.