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Properties

Label 211185l
Number of curves $8$
Conductor $211185$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve([1, -1, 1, -357458, 82346136]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 211185l have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1\)
\(5\)\(1 - T\)
\(13\)\(1 + T\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + T + 2 T^{2}\) 1.2.b
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 211185l do not have complex multiplication.

Modular form 211185.2.a.l

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{4} - q^{5} + 3 q^{8} + q^{10} - 4 q^{11} - q^{13} - q^{16} - 2 q^{17} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 211185l

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
211185.g6 211185l1 \([1, -1, 1, -357458, 82346136]\) \(147281603041/5265\) \(180570794765985\) \([2]\) \(1327104\) \(1.8246\) \(\Gamma_0(N)\)-optimal
211185.g5 211185l2 \([1, -1, 1, -373703, 74464062]\) \(168288035761/27720225\) \(950705234442911025\) \([2, 2]\) \(2654208\) \(2.1712\)  
211185.g7 211185l3 \([1, -1, 1, 682222, 418273242]\) \(1023887723039/2798036865\) \(-95962723741229834385\) \([2]\) \(5308416\) \(2.5178\)  
211185.g4 211185l4 \([1, -1, 1, -1689548, -773992794]\) \(15551989015681/1445900625\) \(49589254512608630625\) \([2, 2]\) \(5308416\) \(2.5178\)  
211185.g8 211185l5 \([1, -1, 1, 1965577, -3667389744]\) \(24487529386319/183539412225\) \(-6294749769487178219025\) \([2]\) \(10616832\) \(2.8644\)  
211185.g2 211185l6 \([1, -1, 1, -26398193, -52197624768]\) \(59319456301170001/594140625\) \(20376912603800390625\) \([2, 2]\) \(10616832\) \(2.8644\)  
211185.g3 211185l7 \([1, -1, 1, -25764638, -54822569844]\) \(-55150149867714721/5950927734375\) \(-204095679124603271484375\) \([2]\) \(21233664\) \(3.2109\)  
211185.g1 211185l8 \([1, -1, 1, -422370068, -3340981629768]\) \(242970740812818720001/24375\) \(835975901694375\) \([2]\) \(21233664\) \(3.2109\)