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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 259210cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
259210.cf4 | 259210cf1 | \([1, 1, 1, -47721101, 126323768899]\) | \(690080604747409/3406760000\) | \(59333066651168936360000\) | \([2]\) | \(63258624\) | \(3.2174\) | \(\Gamma_0(N)\)-optimal |
259210.cf3 | 259210cf2 | \([1, 1, 1, -73642101, -26081342701]\) | \(2535986675931409/1450751712200\) | \(25266689768067035704224200\) | \([2]\) | \(126517248\) | \(3.5640\) | |
259210.cf2 | 259210cf3 | \([1, 1, 1, -274270641, -1657212524437]\) | \(131010595463836369/7704101562500\) | \(134176746085778641601562500\) | \([2]\) | \(189775872\) | \(3.7668\) | |
259210.cf1 | 259210cf4 | \([1, 1, 1, -4324426891, -109457791336937]\) | \(513516182162686336369/1944885031250\) | \(33872651195862646515031250\) | \([2]\) | \(379551744\) | \(4.1133\) |
Rank
sage: E.rank()
The elliptic curves in class 259210cf have rank \(0\).
Complex multiplication
The elliptic curves in class 259210cf do not have complex multiplication.Modular form 259210.2.a.cf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.