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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 15210.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 15210.k1 | 15210n8 | \([1, -1, 0, -8112285, 8895327741]\) | \(16778985534208729/81000\) | \(285018244641000\) | \([2]\) | \(442368\) | \(2.3959\) | |
| 15210.k2 | 15210n7 | \([1, -1, 0, -689805, 30185325]\) | \(10316097499609/5859375000\) | \(20617639224609375000\) | \([2]\) | \(442368\) | \(2.3959\) | |
| 15210.k3 | 15210n6 | \([1, -1, 0, -507285, 138930741]\) | \(4102915888729/9000000\) | \(31668693849000000\) | \([2, 2]\) | \(221184\) | \(2.0494\) | |
| 15210.k4 | 15210n4 | \([1, -1, 0, -438840, -111783294]\) | \(2656166199049/33750\) | \(118757601933750\) | \([2]\) | \(147456\) | \(1.8466\) | |
| 15210.k5 | 15210n5 | \([1, -1, 0, -104220, 11180430]\) | \(35578826569/5314410\) | \(18700047030896010\) | \([2]\) | \(147456\) | \(1.8466\) | |
| 15210.k6 | 15210n2 | \([1, -1, 0, -28170, -1641600]\) | \(702595369/72900\) | \(256516420176900\) | \([2, 2]\) | \(73728\) | \(1.5001\) | |
| 15210.k7 | 15210n3 | \([1, -1, 0, -20565, 3719925]\) | \(-273359449/1536000\) | \(-5404790416896000\) | \([2]\) | \(110592\) | \(1.7028\) | |
| 15210.k8 | 15210n1 | \([1, -1, 0, 2250, -126684]\) | \(357911/2160\) | \(-7600486523760\) | \([2]\) | \(36864\) | \(1.1535\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 15210.k have rank \(0\).
Complex multiplication
The elliptic curves in class 15210.k do not have complex multiplication.Modular form 15210.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
