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Properties

Label 57330.dn
Number of curves $8$
Conductor $57330$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("dn1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 57330.dn have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1 + T\)
\(7\)\(1\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(17\) \( 1 + 6 T + 17 T^{2}\) 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 57330.dn do not have complex multiplication.

Modular form 57330.2.a.dn

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} - q^{13} + q^{16} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 12 & 2 & 3 & 6 & 12 & 4 \\ 4 & 1 & 3 & 2 & 12 & 6 & 12 & 4 \\ 12 & 3 & 1 & 6 & 4 & 2 & 4 & 12 \\ 2 & 2 & 6 & 1 & 6 & 3 & 6 & 2 \\ 3 & 12 & 4 & 6 & 1 & 2 & 4 & 12 \\ 6 & 6 & 2 & 3 & 2 & 1 & 2 & 6 \\ 12 & 12 & 4 & 6 & 4 & 2 & 1 & 3 \\ 4 & 4 & 12 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 57330.dn

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
57330.dn1 57330du7 \([1, -1, 1, -999164228, -12154087454169]\) \(1286229821345376481036009/247265484375000000\) \(21207001452029859375000000\) \([2]\) \(31850496\) \(3.8602\)  
57330.dn2 57330du8 \([1, -1, 1, -439482308, 3434973362727]\) \(109454124781830273937129/3914078300576808000\) \(335695313130744884721768000\) \([2]\) \(31850496\) \(3.8602\)  
57330.dn3 57330du5 \([1, -1, 1, -435638993, 3499868653341]\) \(106607603143751752938169/5290068420\) \(453708648207998820\) \([2]\) \(10616832\) \(3.3109\)  
57330.dn4 57330du6 \([1, -1, 1, -69042308, -147329613273]\) \(424378956393532177129/136231857216000000\) \(11684077950042179136000000\) \([2, 2]\) \(15925248\) \(3.5136\)  
57330.dn5 57330du4 \([1, -1, 1, -30324713, 41480122317]\) \(35958207000163259449/12145729518877500\) \(1041692107549319449177500\) \([2]\) \(10616832\) \(3.3109\)  
57330.dn6 57330du2 \([1, -1, 1, -27228893, 54684413781]\) \(26031421522845051769/5797789779600\) \(497253939769736931600\) \([2, 2]\) \(5308416\) \(2.9643\)  
57330.dn7 57330du1 \([1, -1, 1, -1509773, 1054904757]\) \(-4437543642183289/3033210136320\) \(-260146667570047614720\) \([2]\) \(2654208\) \(2.6177\) \(\Gamma_0(N)\)-optimal
57330.dn8 57330du3 \([1, -1, 1, 12242812, -15712746969]\) \(2366200373628880151/2612420149248000\) \(-224057142623242027008000\) \([2]\) \(7962624\) \(3.1670\)