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Properties

Label 12870.bi
Number of curves $2$
Conductor $12870$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bi1")
 
E.isogeny_class()
 

Elliptic curves in class 12870.bi

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12870.bi1 12870bu2 \([1, -1, 1, -149918, 22380077]\) \(-511157582445795481/8504770560\) \(-6199977738240\) \([3]\) \(51840\) \(1.5856\)  
12870.bi2 12870bu1 \([1, -1, 1, -743, 67007]\) \(-62146192681/2610036000\) \(-1902716244000\) \([]\) \(17280\) \(1.0363\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 12870.bi have rank \(1\).

Complex multiplication

The elliptic curves in class 12870.bi do not have complex multiplication.

Modular form 12870.2.a.bi

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} - q^{7} + q^{8} - q^{10} + q^{11} + q^{13} - q^{14} + q^{16} + 5 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.