Properties

Label 221778.b
Number of curves $2$
Conductor $221778$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 221778.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
221778.b1 221778p2 \([1, -1, 0, -76236, -8827804]\) \(-35937/4\) \(-5454128834101476\) \([]\) \(1870128\) \(1.7570\)  
221778.b2 221778p1 \([1, -1, 0, 5904, 15936]\) \(109503/64\) \(-13300725704256\) \([]\) \(623376\) \(1.2077\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 221778.b have rank \(1\).

Complex multiplication

The elliptic curves in class 221778.b do not have complex multiplication.

Modular form 221778.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 3 q^{5} - 4 q^{7} - q^{8} + 3 q^{10} + q^{13} + 4 q^{14} + q^{16} - 3 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.