Properties

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

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Show commands: SageMath
E = EllipticCurve("be1")
 
E.isogeny_class()
 

Elliptic curves in class 22050be

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
22050.bj8 22050be1 \([1, -1, 0, 16308, 2457216]\) \(357911/2160\) \(-2894606583750000\) \([2]\) \(110592\) \(1.6487\) \(\Gamma_0(N)\)-optimal
22050.bj6 22050be2 \([1, -1, 0, -204192, 32224716]\) \(702595369/72900\) \(97692972201562500\) \([2, 2]\) \(221184\) \(1.9953\)  
22050.bj7 22050be3 \([1, -1, 0, -149067, -72457659]\) \(-273359449/1536000\) \(-2058386904000000000\) \([2]\) \(331776\) \(2.1980\)  
22050.bj5 22050be4 \([1, -1, 0, -755442, -217491534]\) \(35578826569/5314410\) \(7121817673493906250\) \([2]\) \(442368\) \(2.3418\)  
22050.bj4 22050be5 \([1, -1, 0, -3180942, 2184414966]\) \(2656166199049/33750\) \(45228227871093750\) \([2]\) \(442368\) \(2.3418\)  
22050.bj3 22050be6 \([1, -1, 0, -3677067, -2707873659]\) \(4102915888729/9000000\) \(12060860765625000000\) \([2, 2]\) \(663552\) \(2.5446\)  
22050.bj1 22050be7 \([1, -1, 0, -58802067, -173540248659]\) \(16778985534208729/81000\) \(108547746890625000\) \([2]\) \(1327104\) \(2.8911\)  
22050.bj2 22050be8 \([1, -1, 0, -5000067, -584458659]\) \(10316097499609/5859375000\) \(7852122894287109375000\) \([2]\) \(1327104\) \(2.8911\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 22050.2.a.be

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