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Properties

Label 340605.b
Number of curves $2$
Conductor $340605$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 340605.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
340605.b1 340605b2 \([1, -1, 1, -189383, -31811548]\) \(-15590912409/78125\) \(-3764116328203125\) \([]\) \(2114448\) \(1.8360\)  
340605.b2 340605b1 \([1, -1, 1, -158, 23666]\) \(-9/5\) \(-240903445005\) \([]\) \(302064\) \(0.86306\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 340605.b have rank \(1\).

Complex multiplication

The elliptic curves in class 340605.b do not have complex multiplication.

Modular form 340605.2.a.b

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.