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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 43350.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
43350.e1 | 43350n4 | \([1, 1, 0, -47258875, -125064996875]\) | \(30949975477232209/478125000\) | \(180324612158203125000\) | \([2]\) | \(5308416\) | \(3.0219\) | |
43350.e2 | 43350n2 | \([1, 1, 0, -3041875, -1832217875]\) | \(8253429989329/936360000\) | \(353147720450625000000\) | \([2, 2]\) | \(2654208\) | \(2.6753\) | |
43350.e3 | 43350n1 | \([1, 1, 0, -729875, 209278125]\) | \(114013572049/15667200\) | \(5908876891200000000\) | \([2]\) | \(1327104\) | \(2.3287\) | \(\Gamma_0(N)\)-optimal |
43350.e4 | 43350n3 | \([1, 1, 0, 4183125, -9208942875]\) | \(21464092074671/109596256200\) | \(-41334174940143403125000\) | \([2]\) | \(5308416\) | \(3.0219\) |
Rank
sage: E.rank()
The elliptic curves in class 43350.e have rank \(1\).
Complex multiplication
The elliptic curves in class 43350.e do not have complex multiplication.Modular form 43350.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.