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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 5586.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
5586.i1 | 5586j3 | \([1, 1, 0, -889424, -323228562]\) | \(661397832743623417/443352042\) | \(52159924389258\) | \([2]\) | \(61440\) | \(1.9483\) | |
5586.i2 | 5586j2 | \([1, 1, 0, -55934, -5002080]\) | \(164503536215257/4178071044\) | \(491545880255556\) | \([2, 2]\) | \(30720\) | \(1.6017\) | |
5586.i3 | 5586j1 | \([1, 1, 0, -7914, 155268]\) | \(466025146777/177366672\) | \(20867011594128\) | \([2]\) | \(15360\) | \(1.2551\) | \(\Gamma_0(N)\)-optimal |
5586.i4 | 5586j4 | \([1, 1, 0, 9236, -15885470]\) | \(740480746823/927484650666\) | \(-109117641666204234\) | \([2]\) | \(61440\) | \(1.9483\) |
Rank
sage: E.rank()
The elliptic curves in class 5586.i have rank \(1\).
Complex multiplication
The elliptic curves in class 5586.i do not have complex multiplication.Modular form 5586.2.a.i
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.