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Properties

Label 257010.u
Number of curves $2$
Conductor $257010$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("u1")
 
E.isogeny_class()
 

Elliptic curves in class 257010.u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
257010.u1 257010u2 \([1, 0, 0, -103020585235, -12726829810852825]\) \(120919335014493587307392149053865541041/4572304411273201003393904611050\) \(4572304411273201003393904611050\) \([]\) \(915069120\) \(4.9691\)  
257010.u2 257010u1 \([1, 0, 0, -1882007485, 31422615242225]\) \(737204115725967331184137434185041/70010817101641406250000000\) \(70010817101641406250000000\) \([7]\) \(130724160\) \(3.9962\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 257010.u have rank \(1\).

Complex multiplication

The elliptic curves in class 257010.u do not have complex multiplication.

Modular form 257010.2.a.u

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + q^{9} + q^{10} - 2 q^{11} + q^{12} - q^{13} + q^{14} + q^{15} + q^{16} + 4 q^{17} + q^{18} - q^{19} + O(q^{20})\) 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.