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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 2057.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2057.e1 | 2057c3 | \([1, -1, 0, -10973, 445176]\) | \(82483294977/17\) | \(30116537\) | \([2]\) | \(1440\) | \(0.82231\) | |
2057.e2 | 2057c2 | \([1, -1, 0, -688, 7035]\) | \(20346417/289\) | \(511981129\) | \([2, 2]\) | \(720\) | \(0.47574\) | |
2057.e3 | 2057c1 | \([1, -1, 0, -83, -104]\) | \(35937/17\) | \(30116537\) | \([2]\) | \(360\) | \(0.12916\) | \(\Gamma_0(N)\)-optimal |
2057.e4 | 2057c4 | \([1, -1, 0, -83, 18530]\) | \(-35937/83521\) | \(-147962546281\) | \([2]\) | \(1440\) | \(0.82231\) |
Rank
sage: E.rank()
The elliptic curves in class 2057.e have rank \(0\).
Complex multiplication
The elliptic curves in class 2057.e do not have complex multiplication.Modular form 2057.2.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.