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Properties

Label 4410bb
Number of curves $8$
Conductor $4410$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve([1, -1, 1, 652, 19527]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 4410bb have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1 + T\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 2 T + 11 T^{2}\) 1.11.ac
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 6 T + 19 T^{2}\) 1.19.g
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 4410bb do not have complex multiplication.

Modular form 4410.2.a.bb

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} - 2 q^{13} + q^{16} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 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 4410bb

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4410.z8 4410bb1 \([1, -1, 1, 652, 19527]\) \(357911/2160\) \(-185254821360\) \([2]\) \(4608\) \(0.84397\) \(\Gamma_0(N)\)-optimal
4410.z6 4410bb2 \([1, -1, 1, -8168, 259431]\) \(702595369/72900\) \(6252350220900\) \([2, 2]\) \(9216\) \(1.1905\)  
4410.z7 4410bb3 \([1, -1, 1, -5963, -578469]\) \(-273359449/1536000\) \(-131736761856000\) \([2]\) \(13824\) \(1.3933\)  
4410.z5 4410bb4 \([1, -1, 1, -30218, -1733889]\) \(35578826569/5314410\) \(455796331103610\) \([2]\) \(18432\) \(1.5371\)  
4410.z4 4410bb5 \([1, -1, 1, -127238, 17500767]\) \(2656166199049/33750\) \(2894606583750\) \([2]\) \(18432\) \(1.5371\)  
4410.z3 4410bb6 \([1, -1, 1, -147083, -21633573]\) \(4102915888729/9000000\) \(771895089000000\) \([2, 2]\) \(27648\) \(1.7399\)  
4410.z1 4410bb7 \([1, -1, 1, -2352083, -1387851573]\) \(16778985534208729/81000\) \(6947055801000\) \([2]\) \(55296\) \(2.0864\)  
4410.z2 4410bb8 \([1, -1, 1, -200003, -4635669]\) \(10316097499609/5859375000\) \(502535865234375000\) \([2]\) \(55296\) \(2.0864\)