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Properties

Label 281775.cf
Number of curves $4$
Conductor $281775$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cf1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 281775.cf have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1 - T\)
\(5\)\(1\)
\(13\)\(1 + T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 281775.cf do not have complex multiplication.

Modular form 281775.2.a.cf

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{3} - q^{4} + q^{6} + 4 q^{7} - 3 q^{8} + q^{9} - q^{12} - q^{13} + 4 q^{14} - q^{16} + q^{18} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 281775.cf

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
281775.cf1 281775cf4 \([1, 0, 1, -112753501, -460841732977]\) \(420339554066191969/244298925\) \(92137221231458203125\) \([2]\) \(28311552\) \(3.1555\)  
281775.cf2 281775cf2 \([1, 0, 1, -7087876, -7113539227]\) \(104413920565969/2472575625\) \(932530699314931640625\) \([2, 2]\) \(14155776\) \(2.8089\)  
281775.cf3 281775cf1 \([1, 0, 1, -982751, 212610773]\) \(278317173889/109245825\) \(41202009982803515625\) \([2]\) \(7077888\) \(2.4623\) \(\Gamma_0(N)\)-optimal
281775.cf4 281775cf3 \([1, 0, 1, 895749, -22234524977]\) \(210751100351/566398828125\) \(-213617043677911376953125\) \([2]\) \(28311552\) \(3.1555\)