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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 150075bi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
150075.bi4 | 150075bi1 | \([1, -1, 0, 9558, -671409]\) | \(8477185319/21880935\) | \(-249237525234375\) | \([2]\) | \(368640\) | \(1.4462\) | \(\Gamma_0(N)\)-optimal |
150075.bi3 | 150075bi2 | \([1, -1, 0, -81567, -7505784]\) | \(5268932332201/900900225\) | \(10261816625390625\) | \([2, 2]\) | \(737280\) | \(1.7928\) | |
150075.bi2 | 150075bi3 | \([1, -1, 0, -375192, 81462591]\) | \(512787603508921/45649063125\) | \(519971359658203125\) | \([2]\) | \(1474560\) | \(2.1393\) | |
150075.bi1 | 150075bi4 | \([1, -1, 0, -1245942, -534967659]\) | \(18778886261717401/732035835\) | \(8338345683046875\) | \([2]\) | \(1474560\) | \(2.1393\) |
Rank
sage: E.rank()
The elliptic curves in class 150075bi have rank \(0\).
Complex multiplication
The elliptic curves in class 150075bi do not have complex multiplication.Modular form 150075.2.a.bi
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.