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SageMath
E = EllipticCurve("dy1")
E.isogeny_class()
Elliptic curves in class 340032.dy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
340032.dy1 | 340032dy3 | \([0, 1, 0, -441089, 112604415]\) | \(36204575259448513/1527466248\) | \(400416112115712\) | \([4]\) | \(2654208\) | \(1.8821\) | |
340032.dy2 | 340032dy2 | \([0, 1, 0, -28929, 1568511]\) | \(10214075575873/1806590016\) | \(473586733154304\) | \([2, 2]\) | \(1327104\) | \(1.5355\) | |
340032.dy3 | 340032dy1 | \([0, 1, 0, -8449, -278785]\) | \(254478514753/21762048\) | \(5704790310912\) | \([2]\) | \(663552\) | \(1.1889\) | \(\Gamma_0(N)\)-optimal |
340032.dy4 | 340032dy4 | \([0, 1, 0, 55551, 9087231]\) | \(72318867421247/177381135624\) | \(-46499400417017856\) | \([2]\) | \(2654208\) | \(1.8821\) |
Rank
sage: E.rank()
The elliptic curves in class 340032.dy have rank \(0\).
Complex multiplication
The elliptic curves in class 340032.dy do not have complex multiplication.Modular form 340032.2.a.dy
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.