Properties

Label 5712c
Number of curves $2$
Conductor $5712$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 5712c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5712.d2 5712c1 \([0, -1, 0, 16, 144]\) \(415292/9639\) \(-9870336\) \([2]\) \(1280\) \(0.022321\) \(\Gamma_0(N)\)-optimal
5712.d1 5712c2 \([0, -1, 0, -344, 2448]\) \(2204605874/127449\) \(261015552\) \([2]\) \(2560\) \(0.36890\)  

Rank

sage: E.rank()
 

The elliptic curves in class 5712c have rank \(2\).

Complex multiplication

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

Modular form 5712.2.a.c

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} - q^{7} + q^{9} - 6 q^{11} - 4 q^{13} + 2 q^{15} + q^{17} - 6 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.