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Properties

Label 6270f
Number of curves $2$
Conductor $6270$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 6270f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6270.b2 6270f1 \([1, 1, 0, 28, 156]\) \(2294744759/10345500\) \(-10345500\) \([2]\) \(1536\) \(0.028443\) \(\Gamma_0(N)\)-optimal
6270.b1 6270f2 \([1, 1, 0, -302, 1674]\) \(3061889942761/372281250\) \(372281250\) \([2]\) \(3072\) \(0.37502\)  

Rank

sage: E.rank()
 

The elliptic curves in class 6270f have rank \(1\).

Complex multiplication

The elliptic curves in class 6270f do not have complex multiplication.

Modular form 6270.2.a.f

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