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Properties

Label 341205.r
Number of curves $4$
Conductor $341205$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("r1")
 
E.isogeny_class()
 

Elliptic curves in class 341205.r

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
341205.r1 341205r4 \([1, 1, 0, -364227, -84758784]\) \(36097320816649/80625\) \(11935393550625\) \([2]\) \(2168320\) \(1.7544\)  
341205.r2 341205r3 \([1, 1, 0, -62697, 4325874]\) \(184122897769/51282015\) \(7591578680236335\) \([2]\) \(2168320\) \(1.7544\)  
341205.r3 341205r2 \([1, 1, 0, -23022, -1300041]\) \(9116230969/416025\) \(61586630721225\) \([2, 2]\) \(1084160\) \(1.4078\)  
341205.r4 341205r1 \([1, 1, 0, 783, -76464]\) \(357911/17415\) \(-2578045006935\) \([2]\) \(542080\) \(1.0612\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 341205.r have rank \(1\).

Complex multiplication

The elliptic curves in class 341205.r do not have complex multiplication.

Modular form 341205.2.a.r

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

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.