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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 83259.m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
83259.m1 | 83259l4 | \([1, -1, 0, -1109016, -448807311]\) | \(347873904937/395307\) | \(171415472642264763\) | \([2]\) | \(1161216\) | \(2.2203\) | |
83259.m2 | 83259l2 | \([1, -1, 0, -87201, -3091608]\) | \(169112377/88209\) | \(38249733564802881\) | \([2, 2]\) | \(580608\) | \(1.8737\) | |
83259.m3 | 83259l1 | \([1, -1, 0, -49356, 4197339]\) | \(30664297/297\) | \(128786981699673\) | \([2]\) | \(290304\) | \(1.5272\) | \(\Gamma_0(N)\)-optimal |
83259.m4 | 83259l3 | \([1, -1, 0, 329094, -24322653]\) | \(9090072503/5845851\) | \(-2534914160794663659\) | \([2]\) | \(1161216\) | \(2.2203\) |
Rank
sage: E.rank()
The elliptic curves in class 83259.m have rank \(1\).
Complex multiplication
The elliptic curves in class 83259.m do not have complex multiplication.Modular form 83259.2.a.m
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.