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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 17325.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 17325.k1 | 17325u5 | \([1, -1, 1, -1016780, 394882472]\) | \(10206027697760497/5557167\) | \(63299605359375\) | \([2]\) | \(163840\) | \(1.9768\) | |
| 17325.k2 | 17325u3 | \([1, -1, 1, -63905, 6109472]\) | \(2533811507137/58110129\) | \(661910688140625\) | \([2, 2]\) | \(81920\) | \(1.6302\) | |
| 17325.k3 | 17325u2 | \([1, -1, 1, -8780, -174778]\) | \(6570725617/2614689\) | \(29782941890625\) | \([2, 2]\) | \(40960\) | \(1.2837\) | |
| 17325.k4 | 17325u1 | \([1, -1, 1, -7655, -255778]\) | \(4354703137/1617\) | \(18418640625\) | \([2]\) | \(20480\) | \(0.93709\) | \(\Gamma_0(N)\)-optimal |
| 17325.k5 | 17325u6 | \([1, -1, 1, 6970, 18866972]\) | \(3288008303/13504609503\) | \(-153825942620109375\) | \([2]\) | \(163840\) | \(1.9768\) | |
| 17325.k6 | 17325u4 | \([1, -1, 1, 28345, -1288528]\) | \(221115865823/190238433\) | \(-2166934650890625\) | \([2]\) | \(81920\) | \(1.6302\) |
Rank
sage: E.rank()
The elliptic curves in class 17325.k have rank \(1\).
Complex multiplication
The elliptic curves in class 17325.k do not have complex multiplication.Modular form 17325.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
