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Properties

Label 278005k
Number of curves $2$
Conductor $278005$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 278005k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
278005.k1 278005k1 \([1, -1, 1, -877, -2596]\) \(15438249/8225\) \(39700504025\) \([2]\) \(202752\) \(0.72507\) \(\Gamma_0(N)\)-optimal
278005.k2 278005k2 \([1, -1, 1, 3348, -22876]\) \(860085351/541205\) \(-2612293164845\) \([2]\) \(405504\) \(1.0716\)  

Rank

sage: E.rank()
 

The elliptic curves in class 278005k have rank \(1\).

Complex multiplication

The elliptic curves in class 278005k do not have complex multiplication.

Modular form 278005.2.a.k

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.