Properties

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

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Show commands: SageMath
E = EllipticCurve("ba1")
 
E.isogeny_class()
 

Elliptic curves in class 51714ba

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51714.s2 51714ba1 \([1, -1, 1, 152375269, -1584967066549]\) \(50611530622079699/169662750916608\) \(-1311608442537419414528065536\) \([2]\) \(23482368\) \(3.8865\) \(\Gamma_0(N)\)-optimal
51714.s1 51714ba2 \([1, -1, 1, -1467428891, -18734805590965]\) \(45204035637810785581/6545053349462016\) \(50597713308513072449741340672\) \([2]\) \(46964736\) \(4.2331\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 51714.2.a.ba

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