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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 1680.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1680.j1 | 1680p7 | \([0, -1, 0, -30732800, 65587209600]\) | \(783736670177727068275201/360150\) | \(1475174400\) | \([4]\) | \(49152\) | \(2.4876\) | |
| 1680.j2 | 1680p5 | \([0, -1, 0, -1920800, 1025280000]\) | \(191342053882402567201/129708022500\) | \(531284060160000\) | \([2, 4]\) | \(24576\) | \(2.1410\) | |
| 1680.j3 | 1680p8 | \([0, -1, 0, -1908800, 1038710400]\) | \(-187778242790732059201/4984939585440150\) | \(-20418312541962854400\) | \([4]\) | \(49152\) | \(2.4876\) | |
| 1680.j4 | 1680p3 | \([0, -1, 0, -241120, -45479168]\) | \(378499465220294881/120530818800\) | \(493694233804800\) | \([2]\) | \(12288\) | \(1.7944\) | |
| 1680.j5 | 1680p4 | \([0, -1, 0, -120800, 15840000]\) | \(47595748626367201/1215506250000\) | \(4978713600000000\) | \([2, 4]\) | \(12288\) | \(1.7944\) | |
| 1680.j6 | 1680p2 | \([0, -1, 0, -17120, -499968]\) | \(135487869158881/51438240000\) | \(210691031040000\) | \([2, 2]\) | \(6144\) | \(1.4479\) | |
| 1680.j7 | 1680p1 | \([0, -1, 0, 3360, -57600]\) | \(1023887723039/928972800\) | \(-3805072588800\) | \([2]\) | \(3072\) | \(1.1013\) | \(\Gamma_0(N)\)-optimal |
| 1680.j8 | 1680p6 | \([0, -1, 0, 20320, 50499072]\) | \(226523624554079/269165039062500\) | \(-1102500000000000000\) | \([4]\) | \(24576\) | \(2.1410\) |
Rank
sage: E.rank()
The elliptic curves in class 1680.j have rank \(0\).
Complex multiplication
The elliptic curves in class 1680.j do not have complex multiplication.Modular form 1680.2.a.j
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
