Properties

Label 303450b
Number of curves $6$
Conductor $303450$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 303450b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
303450.b5 303450b1 [1, 1, 0, 22108350, 38872384500] [2] 63700992 \(\Gamma_0(N)\)-optimal
303450.b4 303450b2 [1, 1, 0, -125859650, 366029632500] [2, 2] 127401984  
303450.b2 303450b3 [1, 1, 0, -1825179650, 30007268392500] [2] 254803968  
303450.b3 303450b4 [1, 1, 0, -794027650, -8334185895500] [2, 2] 254803968  
303450.b6 303450b5 [1, 1, 0, 311397350, -29720843370500] [2] 509607936  
303450.b1 303450b6 [1, 1, 0, -12590140650, -543747958852500] [2] 509607936  

Rank

sage: E.rank()
 

The elliptic curves in class 303450b have rank \(1\).

Complex multiplication

The elliptic curves in class 303450b do not have complex multiplication.

Modular form 303450.2.a.b

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} + q^{4} + q^{6} - q^{7} - q^{8} + q^{9} - 4q^{11} - q^{12} - 6q^{13} + q^{14} + q^{16} - q^{18} + 4q^{19} + O(q^{20}) \)

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.