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SageMath
E = EllipticCurve("ha1")
E.isogeny_class()
Elliptic curves in class 232050ha
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
232050.ha3 | 232050ha1 | \([1, 0, 0, -254338, 49348292]\) | \(116449478628435289/1996001280\) | \(31187520000000\) | \([2]\) | \(1769472\) | \(1.7186\) | \(\Gamma_0(N)\)-optimal |
232050.ha2 | 232050ha2 | \([1, 0, 0, -262338, 46076292]\) | \(127787213284071769/15197834433600\) | \(237466163025000000\) | \([2, 2]\) | \(3538944\) | \(2.0651\) | |
232050.ha4 | 232050ha3 | \([1, 0, 0, 374662, 235265292]\) | \(372239584720800551/1745320379985000\) | \(-27270630937265625000\) | \([2]\) | \(7077888\) | \(2.4117\) | |
232050.ha1 | 232050ha4 | \([1, 0, 0, -1027338, -352488708]\) | \(7674388308884766169/1007648705929320\) | \(15744511030145625000\) | \([2]\) | \(7077888\) | \(2.4117\) |
Rank
sage: E.rank()
The elliptic curves in class 232050ha have rank \(1\).
Complex multiplication
The elliptic curves in class 232050ha do not have complex multiplication.Modular form 232050.2.a.ha
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.