Properties

Label 22050bd
Number of curves $8$
Conductor $22050$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("bd1")
 
E.isogeny_class()
 

Elliptic curves in class 22050bd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
22050.bo7 22050bd1 \([1, -1, 0, -5485167, 4944093741]\) \(13619385906841/6048000\) \(8104898434500000000\) \([2]\) \(884736\) \(2.5868\) \(\Gamma_0(N)\)-optimal
22050.bo6 22050bd2 \([1, -1, 0, -6367167, 3248007741]\) \(21302308926361/8930250000\) \(11967389094691406250000\) \([2, 2]\) \(1769472\) \(2.9334\)  
22050.bo5 22050bd3 \([1, -1, 0, -16234542, -19130371884]\) \(353108405631241/86318776320\) \(115675415850516480000000\) \([2]\) \(2654208\) \(3.1361\)  
22050.bo8 22050bd4 \([1, -1, 0, 21195333, 23837195241]\) \(785793873833639/637994920500\) \(-854974211702943445312500\) \([2]\) \(3538944\) \(3.2799\)  
22050.bo4 22050bd5 \([1, -1, 0, -48041667, -125901267759]\) \(9150443179640281/184570312500\) \(247341871170043945312500\) \([2]\) \(3538944\) \(3.2799\)  
22050.bo2 22050bd6 \([1, -1, 0, -242026542, -1449071107884]\) \(1169975873419524361/108425318400\) \(145300296521217600000000\) \([2, 2]\) \(5308416\) \(3.4827\)  
22050.bo3 22050bd7 \([1, -1, 0, -224386542, -1669271227884]\) \(-932348627918877961/358766164249920\) \(-480780972715070515005000000\) \([2]\) \(10616832\) \(3.8292\)  
22050.bo1 22050bd8 \([1, -1, 0, -3872338542, -92747787595884]\) \(4791901410190533590281/41160000\) \(55158336568125000000\) \([2]\) \(10616832\) \(3.8292\)  

Rank

sage: E.rank()
 

The elliptic curves in class 22050bd have rank \(1\).

Complex multiplication

The elliptic curves in class 22050bd do not have complex multiplication.

Modular form 22050.2.a.bd

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{8} + 2 q^{13} + q^{16} + 6 q^{17} - 8 q^{19} + O(q^{20})\) Copy content Toggle raw display

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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.