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Properties

Label 67158x
Number of curves $2$
Conductor $67158$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("x1")
 
E.isogeny_class()
 

Elliptic curves in class 67158x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
67158.z2 67158x1 \([1, -1, 0, -2767338, -1120258188]\) \(3215014175651328584353/1115930860975816704\) \(813513597651370377216\) \([]\) \(3483648\) \(2.7140\) \(\Gamma_0(N)\)-optimal
67158.z1 67158x2 \([1, -1, 0, -92735658, 343701220788]\) \(120986373702456846135875233/21429653098766238144\) \(15622217109000587606976\) \([3]\) \(10450944\) \(3.2633\)  

Rank

sage: E.rank()
 

The elliptic curves in class 67158x have rank \(1\).

Complex multiplication

The elliptic curves in class 67158x do not have complex multiplication.

Modular form 67158.2.a.x

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.