Properties

Label 69696bn
Number of curves $4$
Conductor $69696$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bn1")
 
E.isogeny_class()
 

Elliptic curves in class 69696bn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
69696.df3 69696bn1 \([0, 0, 0, -384780, 86653424]\) \(18609625/1188\) \(402198088167456768\) \([2]\) \(737280\) \(2.1286\) \(\Gamma_0(N)\)-optimal
69696.df4 69696bn2 \([0, 0, 0, 312180, 365716208]\) \(9938375/176418\) \(-59726416092867330048\) \([2]\) \(1474560\) \(2.4752\)  
69696.df1 69696bn3 \([0, 0, 0, -5611980, -5097474448]\) \(57736239625/255552\) \(86517277632466255872\) \([2]\) \(2211840\) \(2.6779\)  
69696.df2 69696bn4 \([0, 0, 0, -2824140, -10163537296]\) \(-7357983625/127552392\) \(-43182936198304719962112\) \([2]\) \(4423680\) \(3.0245\)  

Rank

sage: E.rank()
 

The elliptic curves in class 69696bn have rank \(1\).

Complex multiplication

The elliptic curves in class 69696bn do not have complex multiplication.

Modular form 69696.2.a.bn

sage: E.q_eigenform(10)
 
\(q - 2 q^{7} - 4 q^{13} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.