Properties

Label 51376t
Number of curves $4$
Conductor $51376$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("t1")
 
E.isogeny_class()
 

Elliptic curves in class 51376t

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51376.o4 51376t1 \([0, 0, 0, 10309, -136214]\) \(6128487/3952\) \(-78133449392128\) \([2]\) \(137088\) \(1.3547\) \(\Gamma_0(N)\)-optimal
51376.o3 51376t2 \([0, 0, 0, -43771, -1120470]\) \(469097433/244036\) \(4824740499963904\) \([2, 2]\) \(274176\) \(1.7013\)  
51376.o2 51376t3 \([0, 0, 0, -395291, 94844490]\) \(345505073913/3388346\) \(66989666172575744\) \([4]\) \(548352\) \(2.0479\)  
51376.o1 51376t4 \([0, 0, 0, -557531, -160077814]\) \(969417177273/1085318\) \(21457398539313152\) \([2]\) \(548352\) \(2.0479\)  

Rank

sage: E.rank()
 

The elliptic curves in class 51376t have rank \(1\).

Complex multiplication

The elliptic curves in class 51376t do not have complex multiplication.

Modular form 51376.2.a.t

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.