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Properties

Label 225264.cm
Number of curves $4$
Conductor $225264$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cm1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 225264.cm have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(13\)\(1 + T\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 225264.cm do not have complex multiplication.

Modular form 225264.2.a.cm

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

Isogeny matrix

Copy content 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

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 225264.cm

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
225264.cm1 225264ce3 \([0, 1, 0, -9007792, -3501439612]\) \(1677865892403172/861235747047\) \(41490016735763725728768\) \([2]\) \(16588800\) \(3.0322\)  
225264.cm2 225264ce2 \([0, 1, 0, -7224452, -7469727780]\) \(3462397543530448/3602520441\) \(43387839479642501376\) \([2, 2]\) \(8294400\) \(2.6856\)  
225264.cm3 225264ce1 \([0, 1, 0, -7222647, -7473648240]\) \(55356847905445888/60021\) \(45179853176016\) \([2]\) \(4147200\) \(2.3390\) \(\Gamma_0(N)\)-optimal
225264.cm4 225264ce4 \([0, 1, 0, -5469992, -11187077628]\) \(-375718260235972/904469833683\) \(-43572818087465174756352\) \([2]\) \(16588800\) \(3.0322\)