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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 36784bi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
36784.h1 | 36784bi1 | \([0, -1, 0, -52917, 5497021]\) | \(-2258403328/480491\) | \(-3486593500983296\) | \([]\) | \(207360\) | \(1.7037\) | \(\Gamma_0(N)\)-optimal |
36784.h2 | 36784bi2 | \([0, -1, 0, 373003, -31813571]\) | \(790939860992/517504691\) | \(-3755176459848298496\) | \([]\) | \(622080\) | \(2.2530\) |
Rank
sage: E.rank()
The elliptic curves in class 36784bi have rank \(2\).
Complex multiplication
The elliptic curves in class 36784bi do not have complex multiplication.Modular form 36784.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.