Properties

Label 825.b
Number of curves $2$
Conductor $825$
CM no
Rank $1$
Graph

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

Elliptic curves in class 825.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
825.b1 825a1 \([0, -1, 1, -23, 53]\) \(-56197120/3267\) \(-81675\) \([]\) \(72\) \(-0.30121\) \(\Gamma_0(N)\)-optimal
825.b2 825a2 \([0, -1, 1, 127, 38]\) \(8990228480/5314683\) \(-132867075\) \([]\) \(216\) \(0.24810\)  

Rank

sage: E.rank()
 

The elliptic curves in class 825.b have rank \(1\).

Complex multiplication

The elliptic curves in class 825.b do not have complex multiplication.

Modular form 825.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{4} + q^{7} + q^{9} - q^{11} + 2 q^{12} + q^{13} + 4 q^{16} + 6 q^{17} - 7 q^{19} + O(q^{20})\) 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.