Properties

Label 68970s
Number of curves $2$
Conductor $68970$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("s1")
 
E.isogeny_class()
 

Elliptic curves in class 68970s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68970.t2 68970s1 \([1, 0, 1, -3465564, -2483418038]\) \(1952140790365739/49863600\) \(117575760484947600\) \([2]\) \(1757184\) \(2.3828\) \(\Gamma_0(N)\)-optimal
68970.t1 68970s2 \([1, 0, 1, -3598664, -2282383798]\) \(2185814337196139/310797325620\) \(732843836314654143420\) \([2]\) \(3514368\) \(2.7294\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 68970.2.a.s

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.