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Properties

Label 250025.a
Number of curves $2$
Conductor $250025$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 250025.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
250025.a1 250025a2 \([1, -1, 1, -1348980, 588739022]\) \(17374804109361438921/482665506294457\) \(7541648535850890625\) \([2]\) \(3732480\) \(2.4016\)  
250025.a2 250025a1 \([1, -1, 1, -1339855, 597280022]\) \(17024594875172176761/13702740137\) \(214105314640625\) \([2]\) \(1866240\) \(2.0550\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 250025.a have rank \(1\).

Complex multiplication

The elliptic curves in class 250025.a do not have complex multiplication.

Modular form 250025.2.a.a

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