Properties

Label 32368w
Number of curves $4$
Conductor $32368$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("w1")
 
E.isogeny_class()
 

Elliptic curves in class 32368w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
32368.r3 32368w1 \([0, 0, 0, -86411, -9619654]\) \(721734273/13328\) \(1317705808412672\) \([2]\) \(110592\) \(1.6957\) \(\Gamma_0(N)\)-optimal
32368.r2 32368w2 \([0, 0, 0, -178891, 14591610]\) \(6403769793/2775556\) \(274412234601938944\) \([2, 2]\) \(221184\) \(2.0423\)  
32368.r4 32368w3 \([0, 0, 0, 607189, 108135130]\) \(250404380127/196003234\) \(-19378346331742806016\) \([4]\) \(442368\) \(2.3889\)  
32368.r1 32368w4 \([0, 0, 0, -2444651, 1470568986]\) \(16342588257633/8185058\) \(809236079591432192\) \([2]\) \(442368\) \(2.3889\)  

Rank

sage: E.rank()
 

The elliptic curves in class 32368w have rank \(1\).

Complex multiplication

The elliptic curves in class 32368w do not have complex multiplication.

Modular form 32368.2.a.w

sage: E.q_eigenform(10)
 
\(q - 2 q^{5} + q^{7} - 3 q^{9} - 2 q^{13} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.