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SageMath
E = EllipticCurve("bp1")
E.isogeny_class()
Elliptic curves in class 35280.bp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
35280.bp1 | 35280dx7 | \([0, 0, 0, -37633323, 88860133978]\) | \(16778985534208729/81000\) | \(28455140560896000\) | \([2]\) | \(1327104\) | \(2.7796\) | |
35280.bp2 | 35280dx8 | \([0, 0, 0, -3200043, 299882842]\) | \(10316097499609/5859375000\) | \(2058386904000000000000\) | \([2]\) | \(1327104\) | \(2.7796\) | |
35280.bp3 | 35280dx6 | \([0, 0, 0, -2353323, 1386901978]\) | \(4102915888729/9000000\) | \(3161682284544000000\) | \([2, 2]\) | \(663552\) | \(2.4330\) | |
35280.bp4 | 35280dx5 | \([0, 0, 0, -2035803, -1118013302]\) | \(2656166199049/33750\) | \(11856308567040000\) | \([2]\) | \(442368\) | \(2.2303\) | |
35280.bp5 | 35280dx4 | \([0, 0, 0, -483483, 111452362]\) | \(35578826569/5314410\) | \(1866941772200386560\) | \([2]\) | \(442368\) | \(2.2303\) | |
35280.bp6 | 35280dx2 | \([0, 0, 0, -130683, -16472918]\) | \(702595369/72900\) | \(25609626504806400\) | \([2, 2]\) | \(221184\) | \(1.8837\) | |
35280.bp7 | 35280dx3 | \([0, 0, 0, -95403, 37117402]\) | \(-273359449/1536000\) | \(-539593776562176000\) | \([2]\) | \(331776\) | \(2.0864\) | |
35280.bp8 | 35280dx1 | \([0, 0, 0, 10437, -1260182]\) | \(357911/2160\) | \(-758803748290560\) | \([2]\) | \(110592\) | \(1.5371\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 35280.bp have rank \(0\).
Complex multiplication
The elliptic curves in class 35280.bp do not have complex multiplication.Modular form 35280.2.a.bp
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.