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Properties

Label 1734.j
Number of curves $6$
Conductor $1734$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("j1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 1734.j have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 1734.j do not have complex multiplication.

Modular form 1734.2.a.j

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + 2 q^{5} - q^{6} + q^{8} + q^{9} + 2 q^{10} + 4 q^{11} - q^{12} - 2 q^{13} - 2 q^{15} + q^{16} + q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 1734.j

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1734.j1 1734i5 \([1, 1, 1, -8018022, -8742084111]\) \(2361739090258884097/5202\) \(125563633938\) \([2]\) \(36864\) \(2.2637\)  
1734.j2 1734i3 \([1, 1, 1, -501132, -136748439]\) \(576615941610337/27060804\) \(653182023745476\) \([2, 2]\) \(18432\) \(1.9171\)  
1734.j3 1734i6 \([1, 1, 1, -475122, -151542927]\) \(-491411892194497/125563633938\) \(-3030800878069216722\) \([2]\) \(36864\) \(2.2637\)  
1734.j4 1734i2 \([1, 1, 1, -32952, -1912599]\) \(163936758817/30338064\) \(732287113126416\) \([2, 2]\) \(9216\) \(1.5705\)  
1734.j5 1734i1 \([1, 1, 1, -9832, 343913]\) \(4354703137/352512\) \(8508782723328\) \([4]\) \(4608\) \(1.2240\) \(\Gamma_0(N)\)-optimal
1734.j6 1734i4 \([1, 1, 1, 65308, -11031127]\) \(1276229915423/2927177028\) \(-70654937488564932\) \([2]\) \(18432\) \(1.9171\)