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Properties

Label 68354i
Number of curves $2$
Conductor $68354$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("i1")
 
E.isogeny_class()
 

Elliptic curves in class 68354i

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
68354.f2 68354i1 \([1, -1, 1, 621237, -226146597]\) \(26515285573583988393471/37407624743081250944\) \(-37407624743081250944\) \([7]\) \(3040352\) \(2.4413\) \(\Gamma_0(N)\)-optimal
68354.f1 68354i2 \([1, -1, 1, -550339173, -4969145682177]\) \(-18433805126765920887235189777569/12739469393917574594\) \(-12739469393917574594\) \([]\) \(21282464\) \(3.4143\)  

Rank

sage: E.rank()
 

The elliptic curves in class 68354i have rank \(1\).

Complex multiplication

The elliptic curves in class 68354i do not have complex multiplication.

Modular form 68354.2.a.i

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

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.