Properties

Label 305184bc
Number of curves $4$
Conductor $305184$
CM no
Rank $1$
Graph

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

Elliptic curves in class 305184bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
305184.bc3 305184bc1 \([0, -1, 0, -9922, 334480]\) \(69934528/9801\) \(15140628081216\) \([2, 2]\) \(655360\) \(1.2553\) \(\Gamma_0(N)\)-optimal
305184.bc1 305184bc2 \([0, -1, 0, -152977, 23080225]\) \(4004529472/99\) \(9787880779776\) \([2]\) \(1310720\) \(1.6018\)  
305184.bc4 305184bc3 \([0, -1, 0, 16088, 1770232]\) \(37259704/131769\) \(-1628458664735232\) \([2]\) \(1310720\) \(1.6018\)  
305184.bc2 305184bc4 \([0, -1, 0, -41712, -2933532]\) \(649461896/72171\) \(891920636057088\) \([2]\) \(1310720\) \(1.6018\)  

Rank

sage: E.rank()
 

The elliptic curves in class 305184bc have rank \(1\).

Complex multiplication

The elliptic curves in class 305184bc do not have complex multiplication.

Modular form 305184.2.a.bc

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