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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1365.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1365.e1 | 1365e4 | \([1, 0, 1, -564, 5101]\) | \(19790357598649/2998905\) | \(2998905\) | \([2]\) | \(384\) | \(0.25550\) | |
1365.e2 | 1365e3 | \([1, 0, 1, -234, -1343]\) | \(1408317602329/58524375\) | \(58524375\) | \([2]\) | \(384\) | \(0.25550\) | |
1365.e3 | 1365e2 | \([1, 0, 1, -39, 61]\) | \(6321363049/1863225\) | \(1863225\) | \([2, 2]\) | \(192\) | \(-0.091075\) | |
1365.e4 | 1365e1 | \([1, 0, 1, 6, 7]\) | \(30080231/36855\) | \(-36855\) | \([2]\) | \(96\) | \(-0.43765\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1365.e have rank \(0\).
Complex multiplication
The elliptic curves in class 1365.e do not have complex multiplication.Modular form 1365.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.