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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 76440.bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
76440.bk1 | 76440ci4 | \([0, -1, 0, -66248800, -207524286500]\) | \(266912903848829942596/152163375\) | \(18331513759104000\) | \([2]\) | \(4128768\) | \(2.8816\) | |
76440.bk2 | 76440ci2 | \([0, -1, 0, -4141300, -3240297500]\) | \(260798860029250384/196803140625\) | \(5927345328996000000\) | \([2, 2]\) | \(2064384\) | \(2.5350\) | |
76440.bk3 | 76440ci3 | \([0, -1, 0, -3283800, -4621558500]\) | \(-32506165579682596/57814914850875\) | \(-6965111723305567104000\) | \([2]\) | \(4128768\) | \(2.8816\) | |
76440.bk4 | 76440ci1 | \([0, -1, 0, -313175, -27735000]\) | \(1804588288006144/866455078125\) | \(1631001175781250000\) | \([4]\) | \(1032192\) | \(2.1884\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 76440.bk have rank \(1\).
Complex multiplication
The elliptic curves in class 76440.bk do not have complex multiplication.Modular form 76440.2.a.bk
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.