Properties

Label 12480.bj
Number of curves $4$
Conductor $12480$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bj1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 12480.bj

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12480.bj1 12480o3 \([0, -1, 0, -865, 8257]\) \(2186875592/428415\) \(14038302720\) \([4]\) \(6144\) \(0.66378\)  
12480.bj2 12480o2 \([0, -1, 0, -265, -1463]\) \(504358336/38025\) \(155750400\) \([2, 2]\) \(3072\) \(0.31720\)  
12480.bj3 12480o1 \([0, -1, 0, -260, -1530]\) \(30488290624/195\) \(12480\) \([2]\) \(1536\) \(-0.029371\) \(\Gamma_0(N)\)-optimal
12480.bj4 12480o4 \([0, -1, 0, 255, -6975]\) \(55742968/658125\) \(-21565440000\) \([2]\) \(6144\) \(0.66378\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12480.bj have rank \(1\).

Complex multiplication

The elliptic curves in class 12480.bj do not have complex multiplication.

Modular form 12480.2.a.bj

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + q^{9} + q^{13} - q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})\)  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.