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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 76608.by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
76608.by1 | 76608dh3 | \([0, 0, 0, -9739980, -8339164272]\) | \(19804628171203875/5638671302656\) | \(29094305398830674214912\) | \([2]\) | \(5308416\) | \(3.0171\) | |
76608.by2 | 76608dh1 | \([0, 0, 0, -8940300, -10289072624]\) | \(11165451838341046875/572244736\) | \(4050284149997568\) | \([2]\) | \(1769472\) | \(2.4678\) | \(\Gamma_0(N)\)-optimal |
76608.by3 | 76608dh2 | \([0, 0, 0, -8924940, -10326188528]\) | \(-11108001800138902875/79947274872976\) | \(-565857857456138354688\) | \([2]\) | \(3538944\) | \(2.8143\) | |
76608.by4 | 76608dh4 | \([0, 0, 0, 25649460, -55010757744]\) | \(361682234074684125/462672528510976\) | \(-2387288622021093781143552\) | \([2]\) | \(10616832\) | \(3.3636\) |
Rank
sage: E.rank()
The elliptic curves in class 76608.by have rank \(1\).
Complex multiplication
The elliptic curves in class 76608.by do not have complex multiplication.Modular form 76608.2.a.by
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.