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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 8568k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8568.l3 | 8568k1 | \([0, 0, 0, -1074, -13547]\) | \(11745974272/357\) | \(4164048\) | \([2]\) | \(2560\) | \(0.36803\) | \(\Gamma_0(N)\)-optimal |
| 8568.l2 | 8568k2 | \([0, 0, 0, -1119, -12350]\) | \(830321872/127449\) | \(23785042176\) | \([2, 2]\) | \(5120\) | \(0.71461\) | |
| 8568.l1 | 8568k3 | \([0, 0, 0, -4899, 119950]\) | \(17418812548/1753941\) | \(1309309940736\) | \([2]\) | \(10240\) | \(1.0612\) | |
| 8568.l4 | 8568k4 | \([0, 0, 0, 1941, -68042]\) | \(1083360092/3306177\) | \(-2468047905792\) | \([2]\) | \(10240\) | \(1.0612\) |
Rank
sage: E.rank()
The elliptic curves in class 8568k have rank \(0\).
Complex multiplication
The elliptic curves in class 8568k do not have complex multiplication.Modular form 8568.2.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.
