Properties

Label 10241e
Number of curves $2$
Conductor $10241$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 10241e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10241.f1 10241e1 \([0, -1, 1, -1339, -21617]\) \(-2258403328/480491\) \(-56529285659\) \([]\) \(6912\) \(0.78454\) \(\Gamma_0(N)\)-optimal
10241.f2 10241e2 \([0, -1, 1, 9441, 124452]\) \(790939860992/517504691\) \(-60883909391459\) \([]\) \(20736\) \(1.3338\)  

Rank

sage: E.rank()
 

The elliptic curves in class 10241e have rank \(1\).

Complex multiplication

The elliptic curves in class 10241e do not have complex multiplication.

Modular form 10241.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{4} + 3 q^{5} - 2 q^{9} + q^{11} + 2 q^{12} - 2 q^{13} - 3 q^{15} + 4 q^{16} - q^{19} + O(q^{20})\) 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.