Properties

Label 1728.c
Number of curves 33
Conductor 17281728
CM no
Rank 11
Graph

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Show commands: SageMath
E = EllipticCurve("c1")
 
E.isogeny_class()
 

Elliptic curves in class 1728.c

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1728.c1 1728e3 [0,0,0,7884,357264][0, 0, 0, -7884, -357264] 1167051/512-1167051/512 23776267862016-23776267862016 [][] 34563456 1.27381.2738  
1728.c2 1728e1 [0,0,0,204,1136][0, 0, 0, -204, 1136] 132651/2-132651/2 14155776-14155776 [][] 384384 0.175210.17521 Γ0(N)\Gamma_0(N)-optimal
1728.c3 1728e2 [0,0,0,756,5616][0, 0, 0, 756, 5616] 9261/89261/8 41278242816-41278242816 [][] 11521152 0.724520.72452  

Rank

sage: E.rank()
 

The elliptic curves in class 1728.c have rank 11.

Complex multiplication

The elliptic curves in class 1728.c do not have complex multiplication.

Modular form 1728.2.a.c

sage: E.q_eigenform(10)
 
q3q5q7+3q11+4q132q19+O(q20)q - 3 q^{5} - q^{7} + 3 q^{11} + 4 q^{13} - 2 q^{19} + O(q^{20}) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The i,ji,j entry is the smallest degree of a cyclic isogeny between the ii-th and jj-th curve in the isogeny class, in the LMFDB numbering.

(193913331)\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)

Isogeny graph

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

The vertices are labelled with LMFDB labels.