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Properties

Label 138600u
Number of curves $6$
Conductor $138600$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("u1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 138600u have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 138600u do not have complex multiplication.

Modular form 138600.2.a.u

Copy content sage:E.q_eigenform(10)
 
\(q + q^{7} + q^{11} + 2 q^{13} + 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 138600u

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
138600.eo4 138600u1 \([0, 0, 0, -467850, -123170875]\) \(62140690757632/6237\) \(1136693250000\) \([2]\) \(786432\) \(1.7446\) \(\Gamma_0(N)\)-optimal
138600.eo3 138600u2 \([0, 0, 0, -468975, -122548750]\) \(3911877700432/38900169\) \(113432892804000000\) \([2, 2]\) \(1572864\) \(2.0912\)  
138600.eo2 138600u3 \([0, 0, 0, -833475, 94328750]\) \(5489767279588/2847396321\) \(33212030688144000000\) \([2, 2]\) \(3145728\) \(2.4378\)  
138600.eo5 138600u4 \([0, 0, 0, -122475, -299610250]\) \(-17418812548/3314597517\) \(-38661465438288000000\) \([2]\) \(3145728\) \(2.4378\)  
138600.eo1 138600u5 \([0, 0, 0, -10634475, 13335479750]\) \(5701568801608514/6277868289\) \(146450111445792000000\) \([2]\) \(6291456\) \(2.7843\)  
138600.eo6 138600u6 \([0, 0, 0, 3135525, 733337750]\) \(146142660369886/94532266521\) \(-2205248713401888000000\) \([2]\) \(6291456\) \(2.7843\)