Properties

Label 32487m
Number of curves $6$
Conductor $32487$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("m1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 32487m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
32487.e4 32487m1 \([1, 0, 0, -80557, -8807128]\) \(491411892194497/78897\) \(9282153153\) \([2]\) \(73728\) \(1.3151\) \(\Gamma_0(N)\)-optimal
32487.e3 32487m2 \([1, 0, 0, -80802, -8750925]\) \(495909170514577/6224736609\) \(732334037312241\) \([2, 2]\) \(147456\) \(1.6617\)  
32487.e5 32487m3 \([1, 0, 0, -13917, -22783398]\) \(-2533811507137/1904381781393\) \(-224048612199105057\) \([2]\) \(294912\) \(2.0083\)  
32487.e2 32487m4 \([1, 0, 0, -151607, 8879520]\) \(3275619238041697/1605271262049\) \(188858558708802801\) \([2, 2]\) \(294912\) \(2.0083\)  
32487.e6 32487m5 \([1, 0, 0, 552278, 68146637]\) \(158346567380527343/108665074944153\) \(-12784337402104656297\) \([2]\) \(589824\) \(2.3548\)  
32487.e1 32487m6 \([1, 0, 0, -1988372, 1078244103]\) \(7389727131216686257/6115533215337\) \(719486367251182713\) \([2]\) \(589824\) \(2.3548\)  

Rank

sage: E.rank()
 

The elliptic curves in class 32487m have rank \(1\).

Complex multiplication

The elliptic curves in class 32487m do not have complex multiplication.

Modular form 32487.2.a.m

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} + 2q^{5} - q^{6} + 3q^{8} + q^{9} - 2q^{10} - 4q^{11} - q^{12} - q^{13} + 2q^{15} - q^{16} - q^{17} - q^{18} + 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.