Properties

Label 27744bb
Number of curves $4$
Conductor $27744$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bb1")
 
E.isogeny_class()
 

Elliptic curves in class 27744bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
27744.q3 27744bb1 \([0, 1, 0, -674, -3864]\) \(21952/9\) \(13903239744\) \([2, 2]\) \(18432\) \(0.64381\) \(\Gamma_0(N)\)-optimal
27744.q4 27744bb2 \([0, 1, 0, 2216, -25828]\) \(97336/81\) \(-1001033261568\) \([2]\) \(36864\) \(0.99039\)  
27744.q2 27744bb3 \([0, 1, 0, -5009, 132255]\) \(140608/3\) \(296602447872\) \([2]\) \(36864\) \(0.99039\)  
27744.q1 27744bb4 \([0, 1, 0, -9344, -350664]\) \(7301384/3\) \(37075305984\) \([2]\) \(36864\) \(0.99039\)  

Rank

sage: E.rank()
 

The elliptic curves in class 27744bb have rank \(1\).

Complex multiplication

The elliptic curves in class 27744bb do not have complex multiplication.

Modular form 27744.2.a.bb

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{5} - 4 q^{7} + q^{9} + 4 q^{11} - 2 q^{13} - 2 q^{15} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.