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SageMath
E = EllipticCurve("cw1")
E.isogeny_class()
Elliptic curves in class 86490cw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
86490.cx3 | 86490cw1 | \([1, -1, 1, -2106212, 1176933111]\) | \(1597099875769/186000\) | \(120340174121514000\) | \([4]\) | \(2211840\) | \(2.3037\) | \(\Gamma_0(N)\)-optimal |
86490.cx2 | 86490cw2 | \([1, -1, 1, -2279192, 972401559]\) | \(2023804595449/540562500\) | \(349738631040650062500\) | \([2, 2]\) | \(4423680\) | \(2.6503\) | |
86490.cx4 | 86490cw3 | \([1, -1, 1, 5764378, 6294027471]\) | \(32740359775271/45410156250\) | \(-29379925322635253906250\) | \([2]\) | \(8847360\) | \(2.9969\) | |
86490.cx1 | 86490cw4 | \([1, -1, 1, -13090442, -17441319441]\) | \(383432500775449/18701300250\) | \(12099557679482329562250\) | \([2]\) | \(8847360\) | \(2.9969\) |
Rank
sage: E.rank()
The elliptic curves in class 86490cw have rank \(1\).
Complex multiplication
The elliptic curves in class 86490cw do not have complex multiplication.Modular form 86490.2.a.cw
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 Cremona 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 Cremona labels.