Properties

Label 53312.ba
Number of curves $4$
Conductor $53312$
CM no
Rank $2$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 53312.ba

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
53312.ba1 53312o4 \([0, 0, 0, -284396, 58375856]\) \(82483294977/17\) \(524296650752\) \([2]\) \(147456\) \(1.6360\)  
53312.ba2 53312o2 \([0, 0, 0, -17836, 905520]\) \(20346417/289\) \(8913043062784\) \([2, 2]\) \(73728\) \(1.2895\)  
53312.ba3 53312o1 \([0, 0, 0, -2156, -16464]\) \(35937/17\) \(524296650752\) \([2]\) \(36864\) \(0.94289\) \(\Gamma_0(N)\)-optimal
53312.ba4 53312o3 \([0, 0, 0, -2156, 2442160]\) \(-35937/83521\) \(-2575869445144576\) \([2]\) \(147456\) \(1.6360\)  

Rank

sage: E.rank()
 

The elliptic curves in class 53312.ba have rank \(2\).

Complex multiplication

The elliptic curves in class 53312.ba do not have complex multiplication.

Modular form 53312.2.a.ba

sage: E.q_eigenform(10)
 
\(q - 2 q^{5} - 3 q^{9} - 2 q^{13} - q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.