Properties

Label 107712bz
Number of curves $4$
Conductor $107712$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bz1")
 
E.isogeny_class()
 

Elliptic curves in class 107712bz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
107712.ek3 107712bz1 \([0, 0, 0, -222924, -40510352]\) \(6411014266033/296208\) \(56606230315008\) \([2]\) \(589824\) \(1.7141\) \(\Gamma_0(N)\)-optimal
107712.ek2 107712bz2 \([0, 0, 0, -234444, -36091280]\) \(7457162887153/1370924676\) \(261987785455435776\) \([2, 2]\) \(1179648\) \(2.0607\)  
107712.ek4 107712bz3 \([0, 0, 0, 462516, -209773712]\) \(57258048889007/132611470002\) \(-25342446569116925952\) \([2]\) \(2359296\) \(2.4073\)  
107712.ek1 107712bz4 \([0, 0, 0, -1115724, 420411760]\) \(803760366578833/65593817586\) \(12535173747885735936\) \([2]\) \(2359296\) \(2.4073\)  

Rank

sage: E.rank()
 

The elliptic curves in class 107712bz have rank \(1\).

Complex multiplication

The elliptic curves in class 107712bz do not have complex multiplication.

Modular form 107712.2.a.bz

sage: E.q_eigenform(10)
 
\(q + 2 q^{5} + 4 q^{7} + q^{11} + 2 q^{13} - q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.