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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 16170.bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 16170.bn1 | 16170bn8 | \([1, 1, 1, -260876050, 1621699265717]\) | \(16689299266861680229173649/2396798250\) | \(281980917314250\) | \([2]\) | \(1990656\) | \(3.0954\) | |
| 16170.bn2 | 16170bn7 | \([1, 1, 1, -16733550, 23930589717]\) | \(4404531606962679693649/444872222400201750\) | \(52338772093161335685750\) | \([2]\) | \(1990656\) | \(3.0954\) | |
| 16170.bn3 | 16170bn6 | \([1, 1, 1, -16304800, 25333802717]\) | \(4074571110566294433649/48828650062500\) | \(5744641851203062500\) | \([2, 2]\) | \(995328\) | \(2.7488\) | |
| 16170.bn4 | 16170bn4 | \([1, 1, 1, -3675540, -2708417475]\) | \(46676570542430835889/106752955783320\) | \(12559378494951814680\) | \([2]\) | \(663552\) | \(2.5461\) | |
| 16170.bn5 | 16170bn5 | \([1, 1, 1, -3224740, 2217485885]\) | \(31522423139920199089/164434491947880\) | \(19345553543176134120\) | \([2]\) | \(663552\) | \(2.5461\) | |
| 16170.bn6 | 16170bn3 | \([1, 1, 1, -992300, 417302717]\) | \(-918468938249433649/109183593750000\) | \(-12845340621093750000\) | \([4]\) | \(497664\) | \(2.4022\) | |
| 16170.bn7 | 16170bn2 | \([1, 1, 1, -314140, -8540995]\) | \(29141055407581489/16604321025600\) | \(1953481764340814400\) | \([2, 2]\) | \(331776\) | \(2.1995\) | |
| 16170.bn8 | 16170bn1 | \([1, 1, 1, 77860, -1014595]\) | \(443688652450511/260789760000\) | \(-30681654474240000\) | \([4]\) | \(165888\) | \(1.8529\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 16170.bn have rank \(0\).
Complex multiplication
The elliptic curves in class 16170.bn do not have complex multiplication.Modular form 16170.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 4 & 6 & 12 \\ 4 & 1 & 2 & 3 & 12 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 12 & 3 & 6 & 1 & 4 & 12 & 2 & 4 \\ 3 & 12 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
