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Properties

Label 242550.mm
Number of curves $6$
Conductor $242550$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("mm1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 242550.mm have rank \(1\).

L-function data

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

Complex multiplication

The elliptic curves in class 242550.mm do not have complex multiplication.

Modular form 242550.2.a.mm

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{8} + q^{11} - 2 q^{13} + q^{16} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 242550.mm

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
242550.mm1 242550mm6 \([1, -1, 1, -42832355, -107881387603]\) \(6484907238722641/283593750\) \(380042748083496093750\) \([2]\) \(18874368\) \(3.0271\)  
242550.mm2 242550mm3 \([1, -1, 1, -12954605, 17949803897]\) \(179415687049201/1443420\) \(1934320849590937500\) \([2]\) \(9437184\) \(2.6805\)  
242550.mm3 242550mm4 \([1, -1, 1, -2811605, -1506234103]\) \(1834216913521/329422500\) \(441457656173789062500\) \([2, 2]\) \(9437184\) \(2.6805\)  
242550.mm4 242550mm2 \([1, -1, 1, -827105, 267908897]\) \(46694890801/3920400\) \(5253710949506250000\) \([2, 2]\) \(4718592\) \(2.3339\)  
242550.mm5 242550mm1 \([1, -1, 1, 54895, 19184897]\) \(13651919/126720\) \(-169816919580000000\) \([2]\) \(2359296\) \(1.9873\) \(\Gamma_0(N)\)-optimal
242550.mm6 242550mm5 \([1, -1, 1, 5457145, -8700046603]\) \(13411719834479/32153832150\) \(-43089210293602971093750\) \([2]\) \(18874368\) \(3.0271\)