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Properties

Label 21675.d
Number of curves $2$
Conductor $21675$
CM no
Rank $0$
Graph

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

Elliptic curves in class 21675.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21675.d1 21675r2 \([0, 1, 1, -6100308, -5801426206]\) \(-13549359104/243\) \(-450263343574546875\) \([]\) \(881280\) \(2.5142\)  
21675.d2 21675r1 \([0, 1, 1, 40942, -1629706]\) \(4096/3\) \(-5558806710796875\) \([]\) \(176256\) \(1.7095\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 21675.d have rank \(0\).

Complex multiplication

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

Modular form 21675.2.a.d

sage: E.q_eigenform(10)
 
\(q - 2 q^{2} + q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} + q^{9} - 5 q^{11} + 2 q^{12} + q^{13} + 4 q^{14} - 4 q^{16} - 2 q^{18} - 5 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.