Properties

Label 129360.er
Number of curves $4$
Conductor $129360$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("er1")
 
E.isogeny_class()
 

Elliptic curves in class 129360.er

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
129360.er1 129360ha4 \([0, 1, 0, -450813736, 3684055016564]\) \(21026497979043461623321/161783881875\) \(77962084019043840000\) \([2]\) \(23592960\) \(3.4094\)  
129360.er2 129360ha2 \([0, 1, 0, -28194616, 57475824020]\) \(5143681768032498601/14238434358225\) \(6861363461369090150400\) \([2, 2]\) \(11796480\) \(3.0628\)  
129360.er3 129360ha3 \([0, 1, 0, -17081416, 103257762740]\) \(-1143792273008057401/8897444448004035\) \(-4287592209871776619376640\) \([2]\) \(23592960\) \(3.4094\)  
129360.er4 129360ha1 \([0, 1, 0, -2475496, 101611124]\) \(3481467828171481/2005331497785\) \(966349805088388976640\) \([2]\) \(5898240\) \(2.7162\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 129360.er have rank \(1\).

Complex multiplication

The elliptic curves in class 129360.er do not have complex multiplication.

Modular form 129360.2.a.er

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{5} + q^{9} + q^{11} - 6 q^{13} - q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 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.