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Properties

Label 177450.cr
Number of curves $2$
Conductor $177450$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cr1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 177450.cr have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(5\)\(1\)
\(7\)\(1 + T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 6 T + 11 T^{2}\) 1.11.g
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 177450.cr do not have complex multiplication.

Modular form 177450.2.a.cr

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} - q^{7} - q^{8} + q^{9} - 6 q^{11} + q^{12} + q^{14} + q^{16} + 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 177450.cr

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
177450.cr1 177450fv2 \([1, 0, 1, -52793576, -146933883952]\) \(1726143065560493/9662982966\) \(91096431927999011718750\) \([2]\) \(35481600\) \(3.2479\)  
177450.cr2 177450fv1 \([1, 0, 1, -1459826, -4842063952]\) \(-36495256013/1053197964\) \(-9928877756673585937500\) \([2]\) \(17740800\) \(2.9013\) \(\Gamma_0(N)\)-optimal