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SageMath
E = EllipticCurve("fw1")
E.isogeny_class()
Elliptic curves in class 106722.fw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 106722.fw1 | 106722gi4 | \([1, -1, 1, -754152125, 7971064671903]\) | \(312196988566716625/25367712678\) | \(3854368110192979189821318\) | \([2]\) | \(26542080\) | \(3.7628\) | |
| 106722.fw2 | 106722gi3 | \([1, -1, 1, -43917215, 142287305955]\) | \(-61653281712625/21875235228\) | \(-3323721383004117461387868\) | \([2]\) | \(13271040\) | \(3.4162\) | |
| 106722.fw3 | 106722gi2 | \([1, -1, 1, -19371155, -16453985061]\) | \(5290763640625/2291573592\) | \(348181496979235738062552\) | \([2]\) | \(8847360\) | \(3.2135\) | |
| 106722.fw4 | 106722gi1 | \([1, -1, 1, 4107685, -1906495797]\) | \(50447927375/39517632\) | \(-6004305650435576121792\) | \([2]\) | \(4423680\) | \(2.8669\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 106722.fw have rank \(1\).
Complex multiplication
The elliptic curves in class 106722.fw do not have complex multiplication.Modular form 106722.2.a.fw
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.
