Properties

Label 22050.fn
Number of curves $6$
Conductor $22050$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("fn1")
 
E.isogeny_class()
 

Elliptic curves in class 22050.fn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
22050.fn1 22050eq4 \([1, -1, 1, -14817830, 21958278297]\) \(268498407453697/252\) \(337704101437500\) \([2]\) \(786432\) \(2.5161\)  
22050.fn2 22050eq6 \([1, -1, 1, -10077080, -12191879703]\) \(84448510979617/933897762\) \(1251512319645643781250\) \([2]\) \(1572864\) \(2.8627\)  
22050.fn3 22050eq3 \([1, -1, 1, -1146830, 167586297]\) \(124475734657/63011844\) \(84441897452142562500\) \([2, 2]\) \(786432\) \(2.5161\)  
22050.fn4 22050eq2 \([1, -1, 1, -926330, 343104297]\) \(65597103937/63504\) \(85101433562250000\) \([2, 2]\) \(393216\) \(2.1695\)  
22050.fn5 22050eq1 \([1, -1, 1, -44330, 7944297]\) \(-7189057/16128\) \(-21613062492000000\) \([2]\) \(196608\) \(1.8229\) \(\Gamma_0(N)\)-optimal
22050.fn6 22050eq5 \([1, -1, 1, 4255420, 1291254297]\) \(6359387729183/4218578658\) \(-5653298869219462781250\) \([2]\) \(1572864\) \(2.8627\)  

Rank

sage: E.rank()
 

The elliptic curves in class 22050.fn have rank \(0\).

Complex multiplication

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

Modular form 22050.2.a.fn

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{8} + 4 q^{11} + 6 q^{13} + q^{16} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.