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Properties

Label 12675.b
Number of curves $2$
Conductor $12675$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 12675.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12675.b1 12675u1 \([0, -1, 1, -2654708, 1666116818]\) \(-99897344/27\) \(-559221646623046875\) \([]\) \(486720\) \(2.3887\) \(\Gamma_0(N)\)-optimal
12675.b2 12675u2 \([0, -1, 1, 16569042, -3754980682]\) \(24288219136/14348907\) \(-297193311102998654296875\) \([]\) \(2433600\) \(3.1934\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12675.b have rank \(1\).

Complex multiplication

The elliptic curves in class 12675.b do not have complex multiplication.

Modular form 12675.2.a.b

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