↑

Properties

Label 58275.m
Number of curves $8$
Conductor $58275$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve([1, -1, 1, -321644255, 2220139708872]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 58275.m have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1\)
\(5\)\(1\)
\(7\)\(1 - T\)
\(37\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + T + 2 T^{2}\) 1.2.b
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 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 58275.m do not have complex multiplication.

Modular form 58275.2.a.m

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{4} + q^{7} + 3 q^{8} + 4 q^{11} + 2 q^{13} - q^{14} - q^{16} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 2 & 4 & 8 & 16 & 8 & 4 \\ 16 & 1 & 8 & 4 & 2 & 4 & 8 & 16 \\ 2 & 8 & 1 & 2 & 4 & 8 & 4 & 2 \\ 4 & 4 & 2 & 1 & 2 & 4 & 2 & 4 \\ 8 & 2 & 4 & 2 & 1 & 2 & 4 & 8 \\ 16 & 4 & 8 & 4 & 2 & 1 & 8 & 16 \\ 8 & 8 & 4 & 2 & 4 & 8 & 1 & 8 \\ 4 & 16 & 2 & 4 & 8 & 16 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 58275.m

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
58275.m1 58275r8 \([1, -1, 1, -321644255, 2220139708872]\) \(323075148552374741097121/40666351318359375\) \(463215157985687255859375\) \([2]\) \(9437184\) \(3.5633\)  
58275.m2 58275r4 \([1, -1, 1, -125874005, -543534411378]\) \(19363519907006533090561/104895\) \(1194819609375\) \([2]\) \(2359296\) \(2.8702\)  
58275.m3 58275r6 \([1, -1, 1, -21819380, 28419872622]\) \(100856375520095084401/27745356687890625\) \(316036953523004150390625\) \([2, 2]\) \(4718592\) \(3.2167\)  
58275.m4 58275r3 \([1, -1, 1, -7958255, -8284386378]\) \(4893613425692722081/227805440900625\) \(2594846350258681640625\) \([2, 2]\) \(2359296\) \(2.8702\)  
58275.m5 58275r2 \([1, -1, 1, -7867130, -8491240128]\) \(4727429774433470161/11002961025\) \(125330602925390625\) \([2, 2]\) \(1179648\) \(2.5236\)  
58275.m6 58275r1 \([1, -1, 1, -486005, -135806628]\) \(-1114544804970241/55745503695\) \(-634976128025859375\) \([2]\) \(589824\) \(2.1770\) \(\Gamma_0(N)\)-optimal
58275.m7 58275r5 \([1, -1, 1, 4444870, -31751098878]\) \(852615925233775919/38725085979479025\) \(-441102932485003269140625\) \([2]\) \(4718592\) \(3.2167\)  
58275.m8 58275r7 \([1, -1, 1, 56227495, 185606278872]\) \(1725926797674974865599/2305534558266069375\) \(-26261479577749446474609375\) \([2]\) \(9437184\) \(3.5633\)