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SageMath
E = EllipticCurve("ez1")
E.isogeny_class()
Elliptic curves in class 66240.ez
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
66240.ez1 | 66240cw4 | \([0, 0, 0, -63590412, -195180115984]\) | \(148809678420065817601/20700\) | \(3955831603200\) | \([2]\) | \(2359296\) | \(2.7420\) | |
66240.ez2 | 66240cw6 | \([0, 0, 0, -14878092, 18920874224]\) | \(1905890658841300321/293666194803750\) | \(56120483777592360960000\) | \([4]\) | \(4718592\) | \(3.0886\) | |
66240.ez3 | 66240cw3 | \([0, 0, 0, -4078092, -2882165776]\) | \(39248884582600321/3935264062500\) | \(752040673689600000000\) | \([2, 2]\) | \(2359296\) | \(2.7420\) | |
66240.ez4 | 66240cw2 | \([0, 0, 0, -3974412, -3049671184]\) | \(36330796409313601/428490000\) | \(81885714186240000\) | \([2, 2]\) | \(1179648\) | \(2.3955\) | |
66240.ez5 | 66240cw1 | \([0, 0, 0, -241932, -50250256]\) | \(-8194759433281/965779200\) | \(-184563279278899200\) | \([2]\) | \(589824\) | \(2.0489\) | \(\Gamma_0(N)\)-optimal |
66240.ez6 | 66240cw5 | \([0, 0, 0, 5063028, -13964859664]\) | \(75108181893694559/484313964843750\) | \(-92553840000000000000000\) | \([2]\) | \(4718592\) | \(3.0886\) |
Rank
sage: E.rank()
The elliptic curves in class 66240.ez have rank \(0\).
Complex multiplication
The elliptic curves in class 66240.ez do not have complex multiplication.Modular form 66240.2.a.ez
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.