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Properties

Label 112710s
Number of curves $2$
Conductor $112710$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("s1")
 
E.isogeny_class()
 

Elliptic curves in class 112710s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
112710.u1 112710s1 \([1, 1, 0, -204762, 35559936]\) \(39335220262729/23271300\) \(561712609469700\) \([2]\) \(1032192\) \(1.7747\) \(\Gamma_0(N)\)-optimal
112710.u2 112710s2 \([1, 1, 0, -167192, 49062594]\) \(-21413157997609/30812096250\) \(-743729099269016250\) \([2]\) \(2064384\) \(2.1213\)  

Rank

sage: E.rank()
 

The elliptic curves in class 112710s have rank \(1\).

Complex multiplication

The elliptic curves in class 112710s do not have complex multiplication.

Modular form 112710.2.a.s

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + 4 q^{7} - q^{8} + q^{9} - q^{10} - 2 q^{11} - q^{12} + q^{13} - 4 q^{14} - q^{15} + q^{16} - q^{18} - 4 q^{19} + O(q^{20})\) 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.