Properties

Label 5712.b
Number of curves $2$
Conductor $5712$
CM no
Rank $1$
Graph

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

Elliptic curves in class 5712.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5712.b1 5712n1 \([0, -1, 0, -2997, -63639]\) \(-11632923639808/318495051\) \(-81534733056\) \([]\) \(6912\) \(0.87493\) \(\Gamma_0(N)\)-optimal
5712.b2 5712n2 \([0, -1, 0, 13323, -265191]\) \(1021544365555712/705905647251\) \(-180711845696256\) \([]\) \(20736\) \(1.4242\)  

Rank

sage: E.rank()
 

The elliptic curves in class 5712.b have rank \(1\).

Complex multiplication

The elliptic curves in class 5712.b do not have complex multiplication.

Modular form 5712.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{3} - 3 q^{5} - q^{7} + q^{9} + 3 q^{11} - q^{13} + 3 q^{15} + q^{17} - 5 q^{19} + O(q^{20})\) 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.