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Properties

Label 167310.ef
Number of curves $4$
Conductor $167310$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ef1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 167310.ef have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1 + T\)
\(11\)\(1 + T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 167310.ef do not have complex multiplication.

Modular form 167310.2.a.ef

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} + 4 q^{7} + q^{8} - q^{10} - q^{11} + 4 q^{14} + q^{16} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 167310.ef

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
167310.ef1 167310bx4 \([1, -1, 1, -5398519082963, 4827921458397824531]\) \(4944928228995290413834018379264689/189679641808585500000\) \(667434056202674984360065500000\) \([2]\) \(3251404800\) \(5.6629\)  
167310.ef2 167310bx3 \([1, -1, 1, -337391582963, 75443782308824531]\) \(-1207087636168285491836819264689/236446260657750000000000\) \(-831993804501237568797750000000000\) \([2]\) \(1625702400\) \(5.3163\)  
167310.ef3 167310bx2 \([1, -1, 1, -67188001523, 6509984371092947]\) \(9532597152396244075685450929/313550122650789880627200\) \(1103302537838251674178894766899200\) \([2]\) \(1083801600\) \(5.1136\)  
167310.ef4 167310bx1 \([1, -1, 1, 1342174477, 350738860846547]\) \(75991146714893572533071/15147028085515223040000\) \(-53298510573598465542523453440000\) \([2]\) \(541900800\) \(4.7670\) \(\Gamma_0(N)\)-optimal