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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 327990.bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
327990.bf1 | 327990bf4 | \([1, 1, 1, -733837695, -7651781299995]\) | \(73474353581350183614361/576510977802240\) | \(342922174409285678039040\) | \([2]\) | \(108864000\) | \(3.6890\) | |
327990.bf2 | 327990bf3 | \([1, 1, 1, -44890495, -124895350555]\) | \(-16818951115904497561/1592332281446400\) | \(-947156375785454331494400\) | \([2]\) | \(54432000\) | \(3.3425\) | |
327990.bf3 | 327990bf2 | \([1, 1, 1, -13458120, 709765545]\) | \(453198971846635561/261896250564000\) | \(155781997517926603044000\) | \([2]\) | \(36288000\) | \(3.1397\) | |
327990.bf4 | 327990bf1 | \([1, 1, 1, 3361880, 90789545]\) | \(7064514799444439/4094064000000\) | \(-2435244744866544000000\) | \([2]\) | \(18144000\) | \(2.7932\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 327990.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 327990.bf do not have complex multiplication.Modular form 327990.2.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.