Properties

Label 68970.j
Number of curves $4$
Conductor $68970$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("j1")
 
E.isogeny_class()
 

Elliptic curves in class 68970.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68970.j1 68970o4 \([1, 1, 0, -59590082, -177080195376]\) \(13209596798923694545921/92340\) \(163585942740\) \([2]\) \(5529600\) \(2.6864\)  
68970.j2 68970o3 \([1, 1, 0, -3770362, -2696245064]\) \(3345930611358906241/165622259047500\) \(293409934860448147500\) \([2]\) \(5529600\) \(2.6864\)  
68970.j3 68970o2 \([1, 1, 0, -3724382, -2768038236]\) \(3225005357698077121/8526675600\) \(15105525952611600\) \([2, 2]\) \(2764800\) \(2.3399\)  
68970.j4 68970o1 \([1, 1, 0, -229902, -44440524]\) \(-758575480593601/40535043840\) \(-71810302800234240\) \([2]\) \(1382400\) \(1.9933\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 68970.j have rank \(1\).

Complex multiplication

The elliptic curves in class 68970.j do not have complex multiplication.

Modular form 68970.2.a.j

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.