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SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 26010.bl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 26010.bl1 | 26010bm7 | \([1, -1, 1, -13872488, 19890942531]\) | \(16778985534208729/81000\) | \(1425299311881000\) | \([2]\) | \(884736\) | \(2.5301\) | |
| 26010.bl2 | 26010bm8 | \([1, -1, 1, -1179608, 67410627]\) | \(10316097499609/5859375000\) | \(103103248833984375000\) | \([2]\) | \(884736\) | \(2.5301\) | |
| 26010.bl3 | 26010bm6 | \([1, -1, 1, -867488, 310614531]\) | \(4102915888729/9000000\) | \(158366590209000000\) | \([2, 2]\) | \(442368\) | \(2.1835\) | |
| 26010.bl4 | 26010bm5 | \([1, -1, 1, -750443, -250031019]\) | \(2656166199049/33750\) | \(593874713283750\) | \([2]\) | \(294912\) | \(1.9808\) | |
| 26010.bl5 | 26010bm4 | \([1, -1, 1, -178223, 24988317]\) | \(35578826569/5314410\) | \(93513887852512410\) | \([2]\) | \(294912\) | \(1.9808\) | |
| 26010.bl6 | 26010bm2 | \([1, -1, 1, -48173, -3674703]\) | \(702595369/72900\) | \(1282769380692900\) | \([2, 2]\) | \(147456\) | \(1.6342\) | |
| 26010.bl7 | 26010bm3 | \([1, -1, 1, -35168, 8315907]\) | \(-273359449/1536000\) | \(-27027898062336000\) | \([2]\) | \(221184\) | \(1.8369\) | |
| 26010.bl8 | 26010bm1 | \([1, -1, 1, 3847, -282999]\) | \(357911/2160\) | \(-38007981650160\) | \([2]\) | \(73728\) | \(1.2876\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 26010.bl have rank \(1\).
Complex multiplication
The elliptic curves in class 26010.bl do not have complex multiplication.Modular form 26010.2.a.bl
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.
