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SageMath
E = EllipticCurve("q1")
E.isogeny_class()
Elliptic curves in class 80586.q
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
80586.q1 | 80586p4 | \([1, -1, 0, -6106487481, 183670297731949]\) | \(19499096390516434897995817/15393430272\) | \(19880122129322957568\) | \([2]\) | \(58982400\) | \(3.9076\) | |
80586.q2 | 80586p2 | \([1, -1, 0, -381658041, 2869879323757]\) | \(4760617885089919932457/133756441657344\) | \(172742160047877049614336\) | \([2, 2]\) | \(29491200\) | \(3.5610\) | |
80586.q3 | 80586p3 | \([1, -1, 0, -366324921, 3111023301997]\) | \(-4209586785160189454377/801182513521564416\) | \(-1034701553536009834034191104\) | \([2]\) | \(58982400\) | \(3.9076\) | |
80586.q4 | 80586p1 | \([1, -1, 0, -24814521, 41038003309]\) | \(1308451928740468777/194033737531392\) | \(250588356927145899982848\) | \([2]\) | \(14745600\) | \(3.2144\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 80586.q have rank \(0\).
Complex multiplication
The elliptic curves in class 80586.q do not have complex multiplication.Modular form 80586.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.