Properties

Label 3315.a
Number of curves $4$
Conductor $3315$
CM no
Rank $2$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 3315.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3315.a1 3315b3 \([1, 1, 1, -1046, -13456]\) \(126574061279329/16286595\) \(16286595\) \([2]\) \(1792\) \(0.40495\)  
3315.a2 3315b2 \([1, 1, 1, -71, -196]\) \(39616946929/10989225\) \(10989225\) \([2, 2]\) \(896\) \(0.058381\)  
3315.a3 3315b1 \([1, 1, 1, -26, 38]\) \(1948441249/89505\) \(89505\) \([2]\) \(448\) \(-0.28819\) \(\Gamma_0(N)\)-optimal
3315.a4 3315b4 \([1, 1, 1, 184, -1012]\) \(688699320191/910381875\) \(-910381875\) \([2]\) \(1792\) \(0.40495\)  

Rank

sage: E.rank()
 

The elliptic curves in class 3315.a have rank \(2\).

Complex multiplication

The elliptic curves in class 3315.a do not have complex multiplication.

Modular form 3315.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} - 4 q^{7} + 3 q^{8} + q^{9} + q^{10} - 4 q^{11} + q^{12} + q^{13} + 4 q^{14} + q^{15} - q^{16} - q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 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.