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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 47040fk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
47040.dk4 | 47040fk1 | \([0, -1, 0, -78407660, -263920143150]\) | \(7079962908642659949376/100085966990454375\) | \(753600891549437872920000\) | \([2]\) | \(10321920\) | \(3.3864\) | \(\Gamma_0(N)\)-optimal |
47040.dk2 | 47040fk2 | \([0, -1, 0, -1250235065, -17014724166663]\) | \(448487713888272974160064/91549016015625\) | \(44116583158670400000000\) | \([2, 2]\) | \(20643840\) | \(3.7329\) | |
47040.dk3 | 47040fk3 | \([0, -1, 0, -1245948545, -17137192614975]\) | \(-55486311952875723077768/801237030029296875\) | \(-3088866847815000000000000000\) | \([2]\) | \(41287680\) | \(4.0795\) | |
47040.dk1 | 47040fk4 | \([0, -1, 0, -20003760065, -1088962462461663]\) | \(229625675762164624948320008/9568125\) | \(36886293319680000\) | \([2]\) | \(41287680\) | \(4.0795\) |
Rank
sage: E.rank()
The elliptic curves in class 47040fk have rank \(0\).
Complex multiplication
The elliptic curves in class 47040fk do not have complex multiplication.Modular form 47040.2.a.fk
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.