↑

Properties

Label 6825.k
Number of curves $6$
Conductor $6825$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("k1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 6825.k have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1 - T\)
\(5\)\(1\)
\(7\)\(1 + T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 6825.k do not have complex multiplication.

Modular form 6825.2.a.k

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{3} - q^{4} + q^{6} - q^{7} - 3 q^{8} + q^{9} + 4 q^{11} - q^{12} - q^{13} - q^{14} - q^{16} - 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 6825.k

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6825.k1 6825h5 \([1, 0, 1, -3416876, -2431168477]\) \(282352188585428161201/20813369346315\) \(325208896036171875\) \([2]\) \(147456\) \(2.4106\)  
6825.k2 6825h4 \([1, 0, 1, -1171251, 487791523]\) \(11372424889583066401/50586128775\) \(790408262109375\) \([2]\) \(73728\) \(2.0640\)  
6825.k3 6825h3 \([1, 0, 1, -227501, -32758477]\) \(83339496416030401/18593645841225\) \(290525716269140625\) \([2, 2]\) \(73728\) \(2.0640\)  
6825.k4 6825h2 \([1, 0, 1, -74376, 7360273]\) \(2912015927948401/184878500625\) \(2888726572265625\) \([2, 2]\) \(36864\) \(1.7174\)  
6825.k5 6825h1 \([1, 0, 1, 3749, 485273]\) \(373092501599/6718359375\) \(-104974365234375\) \([2]\) \(18432\) \(1.3709\) \(\Gamma_0(N)\)-optimal
6825.k6 6825h6 \([1, 0, 1, 511874, -201335977]\) \(949279533867428399/1670570708285115\) \(-26102667316954921875\) \([2]\) \(147456\) \(2.4106\)