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SageMath
E = EllipticCurve("bg1")
E.isogeny_class()
Elliptic curves in class 4950.bg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4950.bg1 | 4950be5 | \([1, -1, 1, -38494130, 91935940997]\) | \(553808571467029327441/12529687500\) | \(142720971679687500\) | \([2]\) | \(294912\) | \(2.8158\) | |
4950.bg2 | 4950be3 | \([1, -1, 1, -2660630, -1666093003]\) | \(182864522286982801/463015182960\) | \(5274032318403750000\) | \([2]\) | \(147456\) | \(2.4692\) | |
4950.bg3 | 4950be4 | \([1, -1, 1, -2408630, 1433506997]\) | \(135670761487282321/643043610000\) | \(7324668620156250000\) | \([2, 2]\) | \(147456\) | \(2.4692\) | |
4950.bg4 | 4950be6 | \([1, -1, 1, -1171130, 2903656997]\) | \(-15595206456730321/310672490129100\) | \(-3538753832876779687500\) | \([2]\) | \(294912\) | \(2.8158\) | |
4950.bg5 | 4950be2 | \([1, -1, 1, -230630, -3973003]\) | \(119102750067601/68309049600\) | \(778082768100000000\) | \([2, 2]\) | \(73728\) | \(2.1226\) | |
4950.bg6 | 4950be1 | \([1, -1, 1, 57370, -517003]\) | \(1833318007919/1070530560\) | \(-12194012160000000\) | \([4]\) | \(36864\) | \(1.7761\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4950.bg have rank \(1\).
Complex multiplication
The elliptic curves in class 4950.bg do not have complex multiplication.Modular form 4950.2.a.bg
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.