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Properties

Label 67270x
Number of curves $2$
Conductor $67270$
CM no
Rank $1$
Graph

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

Elliptic curves in class 67270x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
67270.bc2 67270x1 \([1, -1, 1, 429670127, -230026433919]\) \(9884598436907013225951/5745985122304000000\) \(-5099582947016035201024000000\) \([2]\) \(33177600\) \(4.0060\) \(\Gamma_0(N)\)-optimal
67270.bc1 67270x2 \([1, -1, 1, -1722969873, -1841062209919]\) \(637362635322644797334049/367193472567398848000\) \(325885558542738998195159488000\) \([2]\) \(66355200\) \(4.3526\)  

Rank

sage: E.rank()
 

The elliptic curves in class 67270x have rank \(1\).

Complex multiplication

The elliptic curves in class 67270x do not have complex multiplication.

Modular form 67270.2.a.x

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