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Properties

Label 348480.ms
Number of curves $6$
Conductor $348480$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ms1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 348480.ms have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 348480.ms do not have complex multiplication.

Modular form 348480.2.a.ms

Copy content sage:E.q_eigenform(10)
 
\(q + q^{5} - 2 q^{13} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 348480.ms

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
348480.ms1 348480ms4 \([0, 0, 0, -229998252, -1342563749296]\) \(15897679904620804/2475\) \(209478170920550400\) \([2]\) \(31457280\) \(3.1702\)  
348480.ms2 348480ms5 \([0, 0, 0, -121969452, 508534829264]\) \(1185450336504002/26043266205\) \(4408481429834653715005440\) \([2]\) \(62914560\) \(3.5168\)  
348480.ms3 348480ms3 \([0, 0, 0, -16554252, -14198064496]\) \(5927735656804/2401490025\) \(203256459766039132569600\) \([2, 2]\) \(31457280\) \(3.1702\)  
348480.ms4 348480ms2 \([0, 0, 0, -14376252, -20973386896]\) \(15529488955216/6125625\) \(129614618257090560000\) \([2, 2]\) \(15728640\) \(2.8236\)  
348480.ms5 348480ms1 \([0, 0, 0, -763752, -429401896]\) \(-37256083456/38671875\) \(-51142131572400000000\) \([2]\) \(7864320\) \(2.4771\) \(\Gamma_0(N)\)-optimal
348480.ms6 348480ms6 \([0, 0, 0, 54012948, -103310324656]\) \(102949393183198/86815346805\) \(-14695692974985575012106240\) \([2]\) \(62914560\) \(3.5168\)