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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 176610.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
176610.k1 | 176610dj3 | \([1, 1, 0, -291368672, 1914189162816]\) | \(4599009330619965070321/426202560\) | \(253515222157901760\) | \([2]\) | \(30965760\) | \(3.2206\) | |
176610.k2 | 176610dj6 | \([1, 1, 0, -144260952, -651617326776]\) | \(558190076008175760241/14707244541053400\) | \(8748212040668504226341400\) | \([2]\) | \(61931520\) | \(3.5672\) | |
176610.k3 | 176610dj4 | \([1, 1, 0, -20633952, 21432786624]\) | \(1633364098002912241/611345405160000\) | \(363642504175361736360000\) | \([2, 2]\) | \(30965760\) | \(3.2206\) | |
176610.k4 | 176610dj2 | \([1, 1, 0, -18211872, 29898925056]\) | \(1123051131566043121/341803929600\) | \(203312948535522201600\) | \([2, 2]\) | \(15482880\) | \(2.8741\) | |
176610.k5 | 176610dj1 | \([1, 1, 0, -988192, 594555904]\) | \(-179415687049201/153259868160\) | \(-91162543754953359360\) | \([2]\) | \(7741440\) | \(2.5275\) | \(\Gamma_0(N)\)-optimal |
176610.k6 | 176610dj5 | \([1, 1, 0, 64239768, 152698481976]\) | \(49288727461474020239/45451852884375000\) | \(-27035822078287366509375000\) | \([2]\) | \(61931520\) | \(3.5672\) |
Rank
sage: E.rank()
The elliptic curves in class 176610.k have rank \(0\).
Complex multiplication
The elliptic curves in class 176610.k do not have complex multiplication.Modular form 176610.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.