Properties

Label 27690bc
Number of curves $2$
Conductor $27690$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bc1")
 
E.isogeny_class()
 

Elliptic curves in class 27690bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
27690.bg2 27690bc1 \([1, 0, 0, -319040, 74649600]\) \(-3591362198523471528961/330727587840000000\) \(-330727587840000000\) \([7]\) \(526848\) \(2.1043\) \(\Gamma_0(N)\)-optimal
27690.bg1 27690bc2 \([1, 0, 0, -8462840, -15758963160]\) \(-67030445471226692469644161/68484636225100842737640\) \(-68484636225100842737640\) \([]\) \(3687936\) \(3.0772\)  

Rank

sage: E.rank()
 

The elliptic curves in class 27690bc have rank \(1\).

Complex multiplication

The elliptic curves in class 27690bc do not have complex multiplication.

Modular form 27690.2.a.bc

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

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.