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Properties

Label 2394.j
Number of curves $6$
Conductor $2394$
CM no
Rank $1$
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 2394.j have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(7\)\(1 - T\)
\(19\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 + 6 T + 11 T^{2}\) 1.11.g
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(23\) \( 1 + 6 T + 23 T^{2}\) 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 2394.j do not have complex multiplication.

Modular form 2394.2.a.j

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 2394.j

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2394.j1 2394n6 \([1, -1, 1, -88080395, 318197707931]\) \(103665426767620308239307625/5961940992\) \(4346254983168\) \([6]\) \(165888\) \(2.8115\)  
2394.j2 2394n5 \([1, -1, 1, -5505035, 4972852379]\) \(25309080274342544331625/191933498523648\) \(139919520423739392\) \([6]\) \(82944\) \(2.4650\)  
2394.j3 2394n4 \([1, -1, 1, -1088420, 435877895]\) \(195607431345044517625/752875610010048\) \(548846319697324992\) \([6]\) \(55296\) \(2.2622\)  
2394.j4 2394n3 \([1, -1, 1, -100580, -352249]\) \(154357248921765625/89242711068672\) \(65057936369061888\) \([6]\) \(27648\) \(1.9157\)  
2394.j5 2394n2 \([1, -1, 1, -71465, -6871507]\) \(55369510069623625/3916046302812\) \(2854797754749948\) \([2]\) \(18432\) \(1.7129\)  
2394.j6 2394n1 \([1, -1, 1, -70205, -7142155]\) \(52492168638015625/293197968\) \(213741318672\) \([2]\) \(9216\) \(1.3664\) \(\Gamma_0(N)\)-optimal