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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 82800.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
82800.f1 | 82800do1 | \([0, 0, 0, -10425, 1837375]\) | \(-687518464/7604375\) | \(-1385897343750000\) | \([]\) | \(414720\) | \(1.5874\) | \(\Gamma_0(N)\)-optimal |
82800.f2 | 82800do2 | \([0, 0, 0, 93075, -47428625]\) | \(489277573376/5615234375\) | \(-1023376464843750000\) | \([]\) | \(1244160\) | \(2.1367\) |
Rank
sage: E.rank()
The elliptic curves in class 82800.f have rank \(1\).
Complex multiplication
The elliptic curves in class 82800.f do not have complex multiplication.Modular form 82800.2.a.f
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.