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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 30030.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 30030.k1 | 30030k4 | \([1, 0, 1, -164874, -25603328]\) | \(495651527576335064089/3950098279627500\) | \(3950098279627500\) | \([2]\) | \(262144\) | \(1.8204\) | |
| 30030.k2 | 30030k2 | \([1, 0, 1, -17454, 224656]\) | \(587993951045227609/318187822352400\) | \(318187822352400\) | \([2, 2]\) | \(131072\) | \(1.4738\) | |
| 30030.k3 | 30030k1 | \([1, 0, 1, -13534, 604112]\) | \(274129705558173529/391575824640\) | \(391575824640\) | \([2]\) | \(65536\) | \(1.1272\) | \(\Gamma_0(N)\)-optimal |
| 30030.k4 | 30030k3 | \([1, 0, 1, 67246, 1783136]\) | \(33630425696691897191/20827008330268140\) | \(-20827008330268140\) | \([2]\) | \(262144\) | \(1.8204\) |
Rank
sage: E.rank()
The elliptic curves in class 30030.k have rank \(1\).
Complex multiplication
The elliptic curves in class 30030.k do not have complex multiplication.Modular form 30030.2.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 LMFDB 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 LMFDB labels.
