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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 32718w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
32718.l3 | 32718w1 | \([1, 0, 0, -1561580, 774335376]\) | \(-421130255542777411888321/15402588724729479168\) | \(-15402588724729479168\) | \([10]\) | \(1920000\) | \(2.4530\) | \(\Gamma_0(N)\)-optimal |
32718.l2 | 32718w2 | \([1, 0, 0, -25200620, 48690669456]\) | \(1769935296778757928512038081/2244956575342666752\) | \(2244956575342666752\) | \([10]\) | \(3840000\) | \(2.7996\) | |
32718.l4 | 32718w3 | \([1, 0, 0, 6991300, -40084233984]\) | \(37791795265406275661467199/716004621065775357793008\) | \(-716004621065775357793008\) | \([2]\) | \(9600000\) | \(3.2577\) | |
32718.l1 | 32718w4 | \([1, 0, 0, -141574640, -612508800804]\) | \(313819633022945271834814191361/19542669977582921378655012\) | \(19542669977582921378655012\) | \([2]\) | \(19200000\) | \(3.6043\) |
Rank
sage: E.rank()
The elliptic curves in class 32718w have rank \(1\).
Complex multiplication
The elliptic curves in class 32718w do not have complex multiplication.Modular form 32718.2.a.w
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.