Properties

Label 1470p
Number of curves $6$
Conductor $1470$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("1470.q1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1470p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
1470.q6 1470p1 [1, 0, 0, 489, 5865] [2] 1536 \(\Gamma_0(N)\)-optimal
1470.q5 1470p2 [1, 0, 0, -3431, 59961] [2, 2] 3072  
1470.q4 1470p3 [1, 0, 0, -18131, -889659] [2] 6144  
1470.q2 1470p4 [1, 0, 0, -51451, 4487405] [2, 2] 6144  
1470.q1 1470p5 [1, 0, 0, -823201, 287410955] [2] 12288  
1470.q3 1470p6 [1, 0, 0, -48021, 5112351] [2] 12288  

Rank

sage: E.rank()
 

The elliptic curves in class 1470p have rank \(0\).

Modular form 1470.2.a.q

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} + 4q^{11} + q^{12} + 2q^{13} - q^{15} + q^{16} - 2q^{17} + q^{18} + 4q^{19} + O(q^{20}) \)

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.