Properties

Label 912.b
Number of curves $4$
Conductor $912$
CM no
Rank $1$
Graph

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

Elliptic curves in class 912.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
912.b1 912g3 \([0, -1, 0, -1624, -24656]\) \(115714886617/1539\) \(6303744\) \([2]\) \(384\) \(0.44814\)  
912.b2 912g2 \([0, -1, 0, -104, -336]\) \(30664297/3249\) \(13307904\) \([2, 2]\) \(192\) \(0.10156\)  
912.b3 912g1 \([0, -1, 0, -24, 48]\) \(389017/57\) \(233472\) \([2]\) \(96\) \(-0.24501\) \(\Gamma_0(N)\)-optimal
912.b4 912g4 \([0, -1, 0, 136, -1872]\) \(67419143/390963\) \(-1601384448\) \([4]\) \(384\) \(0.44814\)  

Rank

sage: E.rank()
 

The elliptic curves in class 912.b have rank \(1\).

Complex multiplication

The elliptic curves in class 912.b do not have complex multiplication.

Modular form 912.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} + q^{9} + 6 q^{13} + 2 q^{15} - 6 q^{17} + q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.