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SageMath
E = EllipticCurve("fm1")
E.isogeny_class()
Elliptic curves in class 130050.fm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
130050.fm1 | 130050br5 | \([1, -1, 1, -1804054955, 29493709544297]\) | \(2361739090258884097/5202\) | \(1430248267825031250\) | \([2]\) | \(37748736\) | \(3.6177\) | |
130050.fm2 | 130050br3 | \([1, -1, 1, -112754705, 460849452797]\) | \(576615941610337/27060804\) | \(7440151489225812562500\) | \([2, 2]\) | \(18874368\) | \(3.2711\) | |
130050.fm3 | 130050br6 | \([1, -1, 1, -106902455, 510815963297]\) | \(-491411892194497/125563633938\) | \(-34522716251757171724031250\) | \([2]\) | \(37748736\) | \(3.6177\) | |
130050.fm4 | 130050br2 | \([1, -1, 1, -7414205, 6410535797]\) | \(163936758817/30338064\) | \(8341207897955582250000\) | \([2, 2]\) | \(9437184\) | \(2.9246\) | |
130050.fm5 | 130050br1 | \([1, -1, 1, -2212205, -1173980203]\) | \(4354703137/352512\) | \(96920353207908000000\) | \([2]\) | \(4718592\) | \(2.5780\) | \(\Gamma_0(N)\)-optimal |
130050.fm6 | 130050br4 | \([1, -1, 1, 14694295, 37318218797]\) | \(1276229915423/2927177028\) | \(-804803897330684928562500\) | \([2]\) | \(18874368\) | \(3.2711\) |
Rank
sage: E.rank()
The elliptic curves in class 130050.fm have rank \(1\).
Complex multiplication
The elliptic curves in class 130050.fm do not have complex multiplication.Modular form 130050.2.a.fm
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.