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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 1050.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1050.c1 | 1050c7 | \([1, 1, 0, -48020000, -128100018750]\) | \(783736670177727068275201/360150\) | \(5627343750\) | \([2]\) | \(49152\) | \(2.5991\) | |
1050.c2 | 1050c5 | \([1, 1, 0, -3001250, -2002500000]\) | \(191342053882402567201/129708022500\) | \(2026687851562500\) | \([2, 2]\) | \(24576\) | \(2.2526\) | |
1050.c3 | 1050c8 | \([1, 1, 0, -2982500, -2028731250]\) | \(-187778242790732059201/4984939585440150\) | \(-77889681022502343750\) | \([2]\) | \(49152\) | \(2.5991\) | |
1050.c4 | 1050c4 | \([1, 1, 0, -376750, 88826500]\) | \(378499465220294881/120530818800\) | \(1883294043750000\) | \([2]\) | \(12288\) | \(1.9060\) | |
1050.c5 | 1050c3 | \([1, 1, 0, -188750, -30937500]\) | \(47595748626367201/1215506250000\) | \(18992285156250000\) | \([2, 2]\) | \(12288\) | \(1.9060\) | |
1050.c6 | 1050c2 | \([1, 1, 0, -26750, 976500]\) | \(135487869158881/51438240000\) | \(803722500000000\) | \([2, 2]\) | \(6144\) | \(1.5594\) | |
1050.c7 | 1050c1 | \([1, 1, 0, 5250, 112500]\) | \(1023887723039/928972800\) | \(-14515200000000\) | \([2]\) | \(3072\) | \(1.2129\) | \(\Gamma_0(N)\)-optimal |
1050.c8 | 1050c6 | \([1, 1, 0, 31750, -98631000]\) | \(226523624554079/269165039062500\) | \(-4205703735351562500\) | \([2]\) | \(24576\) | \(2.2526\) |
Rank
sage: E.rank()
The elliptic curves in class 1050.c have rank \(0\).
Complex multiplication
The elliptic curves in class 1050.c do not have complex multiplication.Modular form 1050.2.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 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.