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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 2670e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2670.f4 | 2670e1 | \([1, 0, 0, -1685, -23775]\) | \(529102162437841/64881000000\) | \(64881000000\) | \([6]\) | \(2880\) | \(0.80482\) | \(\Gamma_0(N)\)-optimal |
| 2670.f3 | 2670e2 | \([1, 0, 0, -6685, 185225]\) | \(33039388998357841/4209544161000\) | \(4209544161000\) | \([6]\) | \(5760\) | \(1.1514\) | |
| 2670.f2 | 2670e3 | \([1, 0, 0, -132185, -18508875]\) | \(255429141422627949841/634472100\) | \(634472100\) | \([2]\) | \(8640\) | \(1.3541\) | |
| 2670.f1 | 2670e4 | \([1, 0, 0, -132235, -18494185]\) | \(255719105183305589041/402554845678410\) | \(402554845678410\) | \([2]\) | \(17280\) | \(1.7007\) |
Rank
sage: E.rank()
The elliptic curves in class 2670e have rank \(0\).
Complex multiplication
The elliptic curves in class 2670e do not have complex multiplication.Modular form 2670.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
