Properties

Label 145728w
Number of curves $6$
Conductor $145728$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("w1")
 
E.isogeny_class()
 

Elliptic curves in class 145728w

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
145728.bx6 145728w1 \([0, 0, 0, 17844, -2689904]\) \(3288008303/18259263\) \(-3489399498866688\) \([2]\) \(524288\) \(1.6644\) \(\Gamma_0(N)\)-optimal
145728.bx5 145728w2 \([0, 0, 0, -215436, -34695920]\) \(5786435182177/627352209\) \(119888874140073984\) \([2, 2]\) \(1048576\) \(2.0110\)  
145728.bx4 145728w3 \([0, 0, 0, -811596, 244068496]\) \(309368403125137/44372288367\) \(8479676358863880192\) \([2]\) \(2097152\) \(2.3576\)  
145728.bx2 145728w4 \([0, 0, 0, -3351756, -2361845360]\) \(21790813729717297/304746849\) \(58238029770522624\) \([2, 2]\) \(2097152\) \(2.3576\)  
145728.bx3 145728w5 \([0, 0, 0, -3256716, -2502086384]\) \(-19989223566735457/2584262514273\) \(-493860257242812776448\) \([2]\) \(4194304\) \(2.7041\)  
145728.bx1 145728w6 \([0, 0, 0, -53627916, -151159168496]\) \(89254274298475942657/17457\) \(3336084652032\) \([2]\) \(4194304\) \(2.7041\)  

Rank

sage: E.rank()
 

The elliptic curves in class 145728w have rank \(1\).

Complex multiplication

The elliptic curves in class 145728w do not have complex multiplication.

Modular form 145728.2.a.w

sage: E.q_eigenform(10)
 
\(q - 2 q^{5} - q^{11} + 2 q^{13} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.