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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 232050gk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 232050.gk4 | 232050gk1 | \([1, 0, 0, -8313, -20101383]\) | \(-4066120948681/11168482590720\) | \(-174507540480000000\) | \([4]\) | \(2949120\) | \(1.9873\) | \(\Gamma_0(N)\)-optimal |
| 232050.gk3 | 232050gk2 | \([1, 0, 0, -1160313, -475141383]\) | \(11056793118237203401/159353257190400\) | \(2489894643600000000\) | \([2, 2]\) | \(5898240\) | \(2.3339\) | |
| 232050.gk2 | 232050gk3 | \([1, 0, 0, -2252313, 561166617]\) | \(80870462846141298121/38087635627860000\) | \(595119306685312500000\) | \([2]\) | \(11796480\) | \(2.6805\) | |
| 232050.gk1 | 232050gk4 | \([1, 0, 0, -18500313, -30629401383]\) | \(44816807438220995641801/9512718589920\) | \(148636227967500000\) | \([2]\) | \(11796480\) | \(2.6805\) |
Rank
sage: E.rank()
The elliptic curves in class 232050gk have rank \(1\).
Complex multiplication
The elliptic curves in class 232050gk do not have complex multiplication.Modular form 232050.2.a.gk
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.
