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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 630c
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 630.a7 | 630c1 | \([1, -1, 0, 1890, -24300]\) | \(1023887723039/928972800\) | \(-677221171200\) | \([2]\) | \(1024\) | \(0.95744\) | \(\Gamma_0(N)\)-optimal |
| 630.a6 | 630c2 | \([1, -1, 0, -9630, -210924]\) | \(135487869158881/51438240000\) | \(37498476960000\) | \([2, 2]\) | \(2048\) | \(1.3040\) | |
| 630.a4 | 630c3 | \([1, -1, 0, -135630, -19186524]\) | \(378499465220294881/120530818800\) | \(87866966905200\) | \([2]\) | \(4096\) | \(1.6506\) | |
| 630.a5 | 630c4 | \([1, -1, 0, -67950, 6682500]\) | \(47595748626367201/1215506250000\) | \(886104056250000\) | \([2, 2]\) | \(4096\) | \(1.6506\) | |
| 630.a2 | 630c5 | \([1, -1, 0, -1080450, 432540000]\) | \(191342053882402567201/129708022500\) | \(94557148402500\) | \([2, 2]\) | \(8192\) | \(1.9972\) | |
| 630.a8 | 630c6 | \([1, -1, 0, 11430, 21304296]\) | \(226523624554079/269165039062500\) | \(-196221313476562500\) | \([2]\) | \(8192\) | \(1.9972\) | |
| 630.a1 | 630c7 | \([1, -1, 0, -17287200, 27669604050]\) | \(783736670177727068275201/360150\) | \(262549350\) | \([2]\) | \(16384\) | \(2.3437\) | |
| 630.a3 | 630c8 | \([1, -1, 0, -1073700, 438205950]\) | \(-187778242790732059201/4984939585440150\) | \(-3634020957785869350\) | \([2]\) | \(16384\) | \(2.3437\) |
Rank
sage: E.rank()
The elliptic curves in class 630c have rank \(0\).
Complex multiplication
The elliptic curves in class 630c do not have complex multiplication.Modular form 630.2.a.c
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 2 & 2 \\ 8 & 4 & 8 & 2 & 4 & 1 & 8 & 8 \\ 16 & 8 & 16 & 4 & 2 & 8 & 1 & 4 \\ 16 & 8 & 16 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
