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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 2850.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2850.q1 | 2850r4 | \([1, 1, 1, -186938, -30118969]\) | \(46237740924063961/1806561830400\) | \(28227528600000000\) | \([2]\) | \(20736\) | \(1.9238\) | |
2850.q2 | 2850r2 | \([1, 1, 1, -27563, 1737281]\) | \(148212258825961/1218375000\) | \(19037109375000\) | \([2]\) | \(6912\) | \(1.3745\) | |
2850.q3 | 2850r1 | \([1, 1, 1, -563, 63281]\) | \(-1263214441/110808000\) | \(-1731375000000\) | \([2]\) | \(3456\) | \(1.0279\) | \(\Gamma_0(N)\)-optimal |
2850.q4 | 2850r3 | \([1, 1, 1, 5062, -1702969]\) | \(918046641959/80912056320\) | \(-1264250880000000\) | \([2]\) | \(10368\) | \(1.5772\) |
Rank
sage: E.rank()
The elliptic curves in class 2850.q have rank \(1\).
Complex multiplication
The elliptic curves in class 2850.q do not have complex multiplication.Modular form 2850.2.a.q
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.