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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 44770.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
44770.b1 | 44770j3 | \([1, 0, 1, -638278, 196220616]\) | \(16232905099479601/4052240\) | \(7178790346640\) | \([2]\) | \(414720\) | \(1.8425\) | |
44770.b2 | 44770j4 | \([1, 0, 1, -635858, 197782968]\) | \(-16048965315233521/256572640900\) | \(-454534084285444900\) | \([2]\) | \(829440\) | \(2.1890\) | |
44770.b3 | 44770j1 | \([1, 0, 1, -9078, 181256]\) | \(46694890801/18944000\) | \(33560451584000\) | \([2]\) | \(138240\) | \(1.2931\) | \(\Gamma_0(N)\)-optimal |
44770.b4 | 44770j2 | \([1, 0, 1, 29642, 1327368]\) | \(1625964918479/1369000000\) | \(-2425267009000000\) | \([2]\) | \(276480\) | \(1.6397\) |
Rank
sage: E.rank()
The elliptic curves in class 44770.b have rank \(2\).
Complex multiplication
The elliptic curves in class 44770.b do not have complex multiplication.Modular form 44770.2.a.b
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.