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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 2730k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2730.k3 | 2730k1 | \([1, 0, 1, -138890144, 630009263342]\) | \(296304326013275547793071733369/268420373544960000000\) | \(268420373544960000000\) | \([2]\) | \(430080\) | \(3.2182\) | \(\Gamma_0(N)\)-optimal |
| 2730.k2 | 2730k2 | \([1, 0, 1, -139893664, 620442907886]\) | \(302773487204995438715379645049/8911747415025000000000000\) | \(8911747415025000000000000\) | \([2, 2]\) | \(860160\) | \(3.5647\) | |
| 2730.k1 | 2730k3 | \([1, 0, 1, -330949984, -1440519827218]\) | \(4008766897254067912673785886329/1423480510711669921875000000\) | \(1423480510711669921875000000\) | \([2]\) | \(1720320\) | \(3.9113\) | |
| 2730.k4 | 2730k4 | \([1, 0, 1, 35106336, 2069162907886]\) | \(4784981304203817469820354951/1852343836482910078035000000\) | \(-1852343836482910078035000000\) | \([2]\) | \(1720320\) | \(3.9113\) |
Rank
sage: E.rank()
The elliptic curves in class 2730k have rank \(0\).
Complex multiplication
The elliptic curves in class 2730k do not have complex multiplication.Modular form 2730.2.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.
