Properties

Label 21675x
Number of curves $2$
Conductor $21675$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("x1")
 
E.isogeny_class()
 

Elliptic curves in class 21675x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21675.h2 21675x1 \([1, 0, 0, -873, -700488]\) \(-24389/70227\) \(-211888632270375\) \([2]\) \(92160\) \(1.4279\) \(\Gamma_0(N)\)-optimal
21675.h1 21675x2 \([1, 0, 0, -123698, -16544913]\) \(69375867029/1003833\) \(3028761037747125\) \([2]\) \(184320\) \(1.7744\)  

Rank

sage: E.rank()
 

The elliptic curves in class 21675x have rank \(1\).

Complex multiplication

The elliptic curves in class 21675x do not have complex multiplication.

Modular form 21675.2.a.x

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

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.