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Properties

Label 441.f
Number of curves $6$
Conductor $441$
CM no
Rank $1$
Graph

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Copy content sage:E = EllipticCurve("f1") E.isogeny_class()
 

Elliptic curves in class 441.f

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
441.f1 441c5 \([1, -1, 0, -345753, -78165914]\) \(53297461115137/147\) \(12607619787\) \([2]\) \(1536\) \(1.5965\)  
441.f2 441c3 \([1, -1, 0, -21618, -1216265]\) \(13027640977/21609\) \(1853320108689\) \([2, 2]\) \(768\) \(1.2499\)  
441.f3 441c4 \([1, -1, 0, -17208, 867901]\) \(6570725617/45927\) \(3938980639167\) \([2]\) \(768\) \(1.2499\)  
441.f4 441c6 \([1, -1, 0, -15003, -1979636]\) \(-4354703137/17294403\) \(-1483273860320763\) \([2]\) \(1536\) \(1.5965\)  
441.f5 441c2 \([1, -1, 0, -1773, -5720]\) \(7189057/3969\) \(340405734249\) \([2, 2]\) \(384\) \(0.90332\)  
441.f6 441c1 \([1, -1, 0, 432, -869]\) \(103823/63\) \(-5403265623\) \([2]\) \(192\) \(0.55675\) \(\Gamma_0(N)\)-optimal

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 441.f have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 441.f do not have complex multiplication.

Modular form 441.2.a.f

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{4} - 2 q^{5} - 3 q^{8} - 2 q^{10} - 4 q^{11} + 2 q^{13} - q^{16} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.