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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 106470.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
106470.k1 | 106470t4 | \([1, -1, 0, -16333290, 25352448876]\) | \(136948444639063849/367281893160\) | \(1292370870085018574760\) | \([2]\) | \(8257536\) | \(2.9251\) | |
106470.k2 | 106470t2 | \([1, -1, 0, -1427490, 51343956]\) | \(91422999252649/52587662400\) | \(185042508975574286400\) | \([2, 2]\) | \(4128768\) | \(2.5785\) | |
106470.k3 | 106470t1 | \([1, -1, 0, -940770, -349615980]\) | \(26168974809769/117411840\) | \(413142179467530240\) | \([2]\) | \(2064384\) | \(2.2320\) | \(\Gamma_0(N)\)-optimal |
106470.k4 | 106470t3 | \([1, -1, 0, 5690790, 405834300]\) | \(5792335463322071/3372408585000\) | \(-11866641668011588185000\) | \([2]\) | \(8257536\) | \(2.9251\) |
Rank
sage: E.rank()
The elliptic curves in class 106470.k have rank \(0\).
Complex multiplication
The elliptic curves in class 106470.k do not have complex multiplication.Modular form 106470.2.a.k
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.