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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 266805.bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 266805.bn1 | 266805bn5 | \([1, -1, 1, -162708362312, 25261761097860774]\) | \(3135316978843283198764801/571725\) | \(86867847951903589725\) | \([2]\) | \(353894400\) | \(4.6203\) | |
| 266805.bn2 | 266805bn3 | \([1, -1, 1, -10169273687, 394716838942374]\) | \(765458482133960722801/326869475625\) | \(49664520370302079835525625\) | \([2, 2]\) | \(176947200\) | \(4.2738\) | |
| 266805.bn3 | 266805bn6 | \([1, -1, 1, -10118847542, 398825036805066]\) | \(-754127868744065783521/15825714261328125\) | \(-2404557681023785551825167578125\) | \([2]\) | \(353894400\) | \(4.6203\) | |
| 266805.bn4 | 266805bn4 | \([1, -1, 1, -1357771757, -10154762932674]\) | \(1821931919215868881/761147600816295\) | \(115648701835088762419812935895\) | \([2]\) | \(176947200\) | \(4.2738\) | |
| 266805.bn5 | 266805bn2 | \([1, -1, 1, -638732282, 6103294828656]\) | \(189674274234120481/3859869269025\) | \(586468208974400196299111025\) | \([2, 2]\) | \(88473600\) | \(3.9272\) | |
| 266805.bn6 | 266805bn1 | \([1, -1, 1, 1866523, 285120242124]\) | \(4733169839/231139696095\) | \(-35119345797419500469239695\) | \([2]\) | \(44236800\) | \(3.5806\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 266805.bn have rank \(1\).
Complex multiplication
The elliptic curves in class 266805.bn do not have complex multiplication.Modular form 266805.2.a.bn
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
