Properties

Label 1728.d
Number of curves $3$
Conductor $1728$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1728.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1728.d1 1728bb3 \([0, 0, 0, -7884, 357264]\) \(-1167051/512\) \(-23776267862016\) \([]\) \(3456\) \(1.2738\)  
1728.d2 1728bb1 \([0, 0, 0, -204, -1136]\) \(-132651/2\) \(-14155776\) \([]\) \(384\) \(0.17521\) \(\Gamma_0(N)\)-optimal
1728.d3 1728bb2 \([0, 0, 0, 756, -5616]\) \(9261/8\) \(-41278242816\) \([]\) \(1152\) \(0.72452\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1728.d have rank \(1\).

Complex multiplication

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

Modular form 1728.2.a.d

sage: E.q_eigenform(10)
 
\(q - 3q^{5} + q^{7} - 3q^{11} + 4q^{13} + 2q^{19} + O(q^{20})\)  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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.