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Properties

Label 412090.b
Number of curves $2$
Conductor $412090$
CM no
Rank $1$
Graph

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

Elliptic curves in class 412090.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
412090.b1 412090b1 \([1, -1, 0, -5426710, 4867159576]\) \(-5154200289/20\) \(-68580761514482420\) \([]\) \(19262880\) \(2.4438\) \(\Gamma_0(N)\)-optimal
412090.b2 412090b2 \([1, -1, 0, 37842740, -46176945200]\) \(1747829720511/1280000000\) \(-4389168736926874880000000\) \([]\) \(134840160\) \(3.4167\)  

Rank

sage: E.rank()
 

The elliptic curves in class 412090.b have rank \(1\).

Complex multiplication

The elliptic curves in class 412090.b do not have complex multiplication.

Modular form 412090.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} - 3 q^{3} + q^{4} - q^{5} + 3 q^{6} - q^{8} + 6 q^{9} + q^{10} + 2 q^{11} - 3 q^{12} + 3 q^{15} + q^{16} + 4 q^{17} - 6 q^{18} + 6 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.