Properties

Label 105.a
Number of curves $4$
Conductor $105$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 105.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
105.a1 105a3 \([1, 0, 1, -113, -469]\) \(157551496201/13125\) \(13125\) \([2]\) \(16\) \(-0.16596\)  
105.a2 105a2 \([1, 0, 1, -8, -7]\) \(47045881/11025\) \(11025\) \([2, 2]\) \(8\) \(-0.51254\)  
105.a3 105a1 \([1, 0, 1, -3, 1]\) \(1771561/105\) \(105\) \([2]\) \(4\) \(-0.85911\) \(\Gamma_0(N)\)-optimal
105.a4 105a4 \([1, 0, 1, 17, -37]\) \(590589719/972405\) \(-972405\) \([4]\) \(16\) \(-0.16596\)  

Rank

sage: E.rank()
 

The elliptic curves in class 105.a have rank \(0\).

Complex multiplication

The elliptic curves in class 105.a do not have complex multiplication.

Modular form 105.2.a.a

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} + q^{7} - 3q^{8} + q^{9} + q^{10} - q^{12} - 6q^{13} + q^{14} + q^{15} - q^{16} + 2q^{17} + q^{18} - 8q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.