Properties

Label 12696.g
Number of curves $4$
Conductor $12696$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("g1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 12696.g

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12696.g1 12696o3 \([0, -1, 0, -397984, -81644900]\) \(45989074372/7555707\) \(1145360182034967552\) \([2]\) \(202752\) \(2.1868\)  
12696.g2 12696o2 \([0, -1, 0, -112324, 13308484]\) \(4135597648/385641\) \(14614709317081344\) \([2, 2]\) \(101376\) \(1.8402\)  
12696.g3 12696o1 \([0, -1, 0, -109679, 14017344]\) \(61604313088/621\) \(1470884593104\) \([4]\) \(50688\) \(1.4937\) \(\Gamma_0(N)\)-optimal
12696.g4 12696o4 \([0, -1, 0, 131016, 62852508]\) \(1640689628/12223143\) \(-1852890972548226048\) \([2]\) \(202752\) \(2.1868\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12696.g have rank \(1\).

Complex multiplication

The elliptic curves in class 12696.g do not have complex multiplication.

Modular form 12696.2.a.g

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2q^{5} + 4q^{7} + q^{9} + 4q^{11} - 2q^{13} + 2q^{15} + 2q^{17} + O(q^{20})\)  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.