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Properties

Label 1848.g
Number of curves $6$
Conductor $1848$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("g1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 1848.g have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 1848.g do not have complex multiplication.

Modular form 1848.2.a.g

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - 2 q^{5} - q^{7} + q^{9} - q^{11} - 2 q^{13} - 2 q^{15} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 1848.g

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1848.g1 1848j5 \([0, 1, 0, -47264, -3967008]\) \(5701568801608514/6277868289\) \(12857074255872\) \([2]\) \(6144\) \(1.4303\)  
1848.g2 1848j3 \([0, 1, 0, -3704, -29184]\) \(5489767279588/2847396321\) \(2915733832704\) \([2, 2]\) \(3072\) \(1.0837\)  
1848.g3 1848j2 \([0, 1, 0, -2084, 35616]\) \(3911877700432/38900169\) \(9958443264\) \([2, 4]\) \(1536\) \(0.73716\)  
1848.g4 1848j1 \([0, 1, 0, -2079, 35802]\) \(62140690757632/6237\) \(99792\) \([4]\) \(768\) \(0.39058\) \(\Gamma_0(N)\)-optimal
1848.g5 1848j4 \([0, 1, 0, -544, 88592]\) \(-17418812548/3314597517\) \(-3394147857408\) \([4]\) \(3072\) \(1.0837\)  
1848.g6 1848j6 \([0, 1, 0, 13936, -212640]\) \(146142660369886/94532266521\) \(-193602081835008\) \([2]\) \(6144\) \(1.4303\)