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Properties

Label 80850fw
Number of curves $8$
Conductor $80850$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("fw1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 80850fw have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 80850fw do not have complex multiplication.

Modular form 80850.2.a.fw

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} - q^{11} + q^{12} + 2 q^{13} + q^{16} + 6 q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 80850fw

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
80850.gi7 80850fw1 \([1, 0, 0, -1436338, -301357708]\) \(178272935636041/81841914000\) \(150447177190406250000\) \([4]\) \(2654208\) \(2.5665\) \(\Gamma_0(N)\)-optimal
80850.gi5 80850fw2 \([1, 0, 0, -19296838, -32611002208]\) \(432288716775559561/270140062500\) \(496589190829101562500\) \([2, 2]\) \(5308416\) \(2.9131\)  
80850.gi4 80850fw3 \([1, 0, 0, -58490713, 172164095417]\) \(12038605770121350841/757333463040\) \(1392180071768640000000\) \([4]\) \(7962624\) \(3.1158\)  
80850.gi6 80850fw4 \([1, 0, 0, -15658588, -45283026958]\) \(-230979395175477481/348191894531250\) \(-640069190620422363281250\) \([2]\) \(10616832\) \(3.2596\)  
80850.gi2 80850fw5 \([1, 0, 0, -308703088, -2087684783458]\) \(1769857772964702379561/691787250\) \(1271688721488281250\) \([2]\) \(10616832\) \(3.2596\)  
80850.gi3 80850fw6 \([1, 0, 0, -62018713, 150223463417]\) \(14351050585434661561/3001282273281600\) \(5517154033895421225000000\) \([2, 2]\) \(15925248\) \(3.4624\)  
80850.gi8 80850fw7 \([1, 0, 0, 133638287, 906829082417]\) \(143584693754978072519/276341298967965000\) \(-507988710660658035703125000\) \([2]\) \(31850496\) \(3.8090\)  
80850.gi1 80850fw8 \([1, 0, 0, -314123713, -2010568491583]\) \(1864737106103260904761/129177711985836360\) \(237462947459713467463125000\) \([2]\) \(31850496\) \(3.8090\)