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Properties

Label 2850.z
Number of curves $2$
Conductor $2850$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("z1")
 
E.isogeny_class()
 

Elliptic curves in class 2850.z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2850.z1 2850w2 \([1, 0, 0, -496888, -134878108]\) \(-1389310279182025/267418692\) \(-2611510664062500\) \([]\) \(36000\) \(1.9587\)  
2850.z2 2850w1 \([1, 0, 0, 4772, 7952]\) \(480705753733655/279172334592\) \(-6979308364800\) \([5]\) \(7200\) \(1.1539\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 2850.z have rank \(1\).

Complex multiplication

The elliptic curves in class 2850.z do not have complex multiplication.

Modular form 2850.2.a.z

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

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.