↑

Properties

Label 12495.m
Number of curves $4$
Conductor $12495$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("m1")
 
E.isogeny_class()
 

Elliptic curves in class 12495.m

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
12495.m1 12495j3 \([1, 0, 1, -14579, -663169]\) \(2912566550041/76531875\) \(9003898561875\) \([2]\) \(24576\) \(1.2671\)  
12495.m2 12495j2 \([1, 0, 1, -2084, 21557]\) \(8502154921/3186225\) \(374856185025\) \([2, 2]\) \(12288\) \(0.92054\)  
12495.m3 12495j1 \([1, 0, 1, -1839, 30181]\) \(5841725401/1785\) \(210003465\) \([2]\) \(6144\) \(0.57397\) \(\Gamma_0(N)\)-optimal
12495.m4 12495j4 \([1, 0, 1, 6491, 155327]\) \(257138126279/236782035\) \(-27857169635715\) \([2]\) \(24576\) \(1.2671\)  

Rank

sage: E.rank()
 

The elliptic curves in class 12495.m have rank \(1\).

Complex multiplication

The elliptic curves in class 12495.m do not have complex multiplication.

Modular form 12495.2.a.m

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} - 3 q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} + 2 q^{13} - q^{15} - q^{16} - q^{17} + q^{18} + O(q^{20})\) 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.