Properties

Label 6930.p
Number of curves $4$
Conductor $6930$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("p1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 6930.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6930.p1 6930q3 \([1, -1, 0, -218349, 39325905]\) \(1579250141304807889/41926500\) \(30564418500\) \([6]\) \(41472\) \(1.5260\)  
6930.p2 6930q4 \([1, -1, 0, -218079, 39427803]\) \(-1573398910560073969/8138108343750\) \(-5932680982593750\) \([6]\) \(82944\) \(1.8726\)  
6930.p3 6930q1 \([1, -1, 0, -2889, 46413]\) \(3658671062929/880165440\) \(641640605760\) \([2]\) \(13824\) \(0.97671\) \(\Gamma_0(N)\)-optimal
6930.p4 6930q2 \([1, -1, 0, 6831, 285525]\) \(48351870250991/76871856600\) \(-56039583461400\) \([2]\) \(27648\) \(1.3233\)  

Rank

sage: E.rank()
 

The elliptic curves in class 6930.p have rank \(1\).

Complex multiplication

The elliptic curves in class 6930.p do not have complex multiplication.

Modular form 6930.2.a.p

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - q^{10} + q^{11} - 4 q^{13} - q^{14} + q^{16} + 8 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.