Properties

Label 21840bc
Number of curves $4$
Conductor $21840$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bc1")
 
E.isogeny_class()
 

Elliptic curves in class 21840bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21840.h4 21840bc1 \([0, -1, 0, 104, -464]\) \(30080231/36855\) \(-150958080\) \([2]\) \(6144\) \(0.25550\) \(\Gamma_0(N)\)-optimal
21840.h3 21840bc2 \([0, -1, 0, -616, -3920]\) \(6321363049/1863225\) \(7631769600\) \([2, 2]\) \(12288\) \(0.60207\)  
21840.h2 21840bc3 \([0, -1, 0, -3736, 85936]\) \(1408317602329/58524375\) \(239715840000\) \([2]\) \(24576\) \(0.94865\)  
21840.h1 21840bc4 \([0, -1, 0, -9016, -326480]\) \(19790357598649/2998905\) \(12283514880\) \([2]\) \(24576\) \(0.94865\)  

Rank

sage: E.rank()
 

The elliptic curves in class 21840bc have rank \(1\).

Complex multiplication

The elliptic curves in class 21840bc do not have complex multiplication.

Modular form 21840.2.a.bc

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} - q^{7} + q^{9} + q^{13} + q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.