Properties

Label 51714c
Number of curves $2$
Conductor $51714$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 51714c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51714.j1 51714c1 \([1, -1, 0, -82926, 8880340]\) \(17923019113/735488\) \(2587993811290368\) \([2]\) \(258048\) \(1.7230\) \(\Gamma_0(N)\)-optimal
51714.j2 51714c2 \([1, -1, 0, 38754, 32656612]\) \(1829276567/132066064\) \(-464706638739826704\) \([2]\) \(516096\) \(2.0696\)  

Rank

sage: E.rank()
 

The elliptic curves in class 51714c have rank \(0\).

Complex multiplication

The elliptic curves in class 51714c do not have complex multiplication.

Modular form 51714.2.a.c

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + 2 q^{5} - 2 q^{7} - q^{8} - 2 q^{10} + 2 q^{11} + 2 q^{14} + q^{16} - q^{17} + 4 q^{19} + O(q^{20})\) 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.