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SageMath
E = EllipticCurve("s1")
E.isogeny_class()
Elliptic curves in class 141610s
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
141610.cg4 | 141610s1 | \([1, -1, 1, 32747, 3625437]\) | \(1367631/2800\) | \(-7951330394786800\) | \([2]\) | \(983040\) | \(1.7354\) | \(\Gamma_0(N)\)-optimal |
141610.cg3 | 141610s2 | \([1, -1, 1, -250473, 39084581]\) | \(611960049/122500\) | \(347870704771922500\) | \([2, 2]\) | \(1966080\) | \(2.0820\) | |
141610.cg1 | 141610s3 | \([1, -1, 1, -3790723, 2841546481]\) | \(2121328796049/120050\) | \(340913290676484050\) | \([2]\) | \(3932160\) | \(2.4286\) | |
141610.cg2 | 141610s4 | \([1, -1, 1, -1241743, -497390743]\) | \(74565301329/5468750\) | \(15529942177317968750\) | \([2]\) | \(3932160\) | \(2.4286\) |
Rank
sage: E.rank()
The elliptic curves in class 141610s have rank \(1\).
Complex multiplication
The elliptic curves in class 141610s do not have complex multiplication.Modular form 141610.2.a.s
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 Cremona 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 Cremona labels.