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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 14784.y
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
14784.y1 | 14784bu4 | \([0, -1, 0, -14497, 676417]\) | \(1285429208617/614922\) | \(161198112768\) | \([4]\) | \(24576\) | \(1.1054\) | |
14784.y2 | 14784bu3 | \([0, -1, 0, -8097, -273087]\) | \(223980311017/4278582\) | \(1121604599808\) | \([2]\) | \(24576\) | \(1.1054\) | |
14784.y3 | 14784bu2 | \([0, -1, 0, -1057, 7105]\) | \(498677257/213444\) | \(55953063936\) | \([2, 2]\) | \(12288\) | \(0.75881\) | |
14784.y4 | 14784bu1 | \([0, -1, 0, 223, 705]\) | \(4657463/3696\) | \(-968884224\) | \([2]\) | \(6144\) | \(0.41223\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 14784.y have rank \(1\).
Complex multiplication
The elliptic curves in class 14784.y do not have complex multiplication.Modular form 14784.2.a.y
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.