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Properties

Label 3136b
Number of curves $2$
Conductor $3136$
CM no
Rank $1$
Graph

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

Elliptic curves in class 3136b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3136.h2 3136b1 \([0, -1, 0, -457, -559]\) \(1792\) \(5903156224\) \([]\) \(1344\) \(0.56519\) \(\Gamma_0(N)\)-optimal
3136.h1 3136b2 \([0, -1, 0, -27897, -1784159]\) \(406749952\) \(5903156224\) \([]\) \(4032\) \(1.1145\)  

Rank

sage: E.rank()
 

The elliptic curves in class 3136b have rank \(1\).

Complex multiplication

The elliptic curves in class 3136b do not have complex multiplication.

Modular form 3136.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{3} - 3 q^{5} - 2 q^{9} + 3 q^{11} - 2 q^{13} + 3 q^{15} + 3 q^{17} + q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.