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SageMath
E = EllipticCurve("cj1")
E.isogeny_class()
Elliptic curves in class 84525cj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
84525.cx3 | 84525cj1 | \([1, 0, 1, -343026, -77132177]\) | \(2428257525121/8150625\) | \(14983013759765625\) | \([2]\) | \(589824\) | \(1.9681\) | \(\Gamma_0(N)\)-optimal |
84525.cx2 | 84525cj2 | \([1, 0, 1, -496151, -1488427]\) | \(7347774183121/4251692025\) | \(7815739297644140625\) | \([2, 2]\) | \(1179648\) | \(2.3147\) | |
84525.cx4 | 84525cj3 | \([1, 0, 1, 1984474, -11410927]\) | \(470166844956479/272118787605\) | \(-500226613170947578125\) | \([2]\) | \(2359296\) | \(2.6613\) | |
84525.cx1 | 84525cj4 | \([1, 0, 1, -5426776, 4850246573]\) | \(9614816895690721/34652610405\) | \(63700702524028828125\) | \([2]\) | \(2359296\) | \(2.6613\) |
Rank
sage: E.rank()
The elliptic curves in class 84525cj have rank \(0\).
Complex multiplication
The elliptic curves in class 84525cj do not have complex multiplication.Modular form 84525.2.a.cj
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.