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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 21840k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21840.bk3 | 21840k1 | \([0, 1, 0, -1614796, -790352116]\) | \(1819018058610682173904/4844385\) | \(1240162560\) | \([2]\) | \(172032\) | \(1.8659\) | \(\Gamma_0(N)\)-optimal |
21840.bk2 | 21840k2 | \([0, 1, 0, -1614816, -790331580]\) | \(454771411897393003396/23468066028225\) | \(24031299612902400\) | \([2, 2]\) | \(344064\) | \(2.2125\) | |
21840.bk4 | 21840k3 | \([0, 1, 0, -1526936, -880074636]\) | \(-192245661431796830258/51935513760073125\) | \(-106363932180629760000\) | \([2]\) | \(688128\) | \(2.5590\) | |
21840.bk1 | 21840k4 | \([0, 1, 0, -1703016, -699273900]\) | \(266716694084614489298/51372277695070605\) | \(105210424719504599040\) | \([2]\) | \(688128\) | \(2.5590\) |
Rank
sage: E.rank()
The elliptic curves in class 21840k have rank \(0\).
Complex multiplication
The elliptic curves in class 21840k do not have complex multiplication.Modular form 21840.2.a.k
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.