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Properties

Label 210210cd
Number of curves $4$
Conductor $210210$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cd1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 210210cd have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(7\)\(1\)
\(11\)\(1 + T\)
\(13\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 + 10 T + 29 T^{2}\) 1.29.k
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 210210cd do not have complex multiplication.

Modular form 210210.2.a.cd

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} + q^{8} + q^{9} + q^{10} - q^{11} - q^{12} - q^{13} - q^{15} + q^{16} + q^{18} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 210210cd

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
210210.dk4 210210cd1 \([1, 1, 1, 43239020, 2028095771477]\) \(75991146714893572533071/15147028085515223040000\) \(-1782032707232780475432960000\) \([2]\) \(116121600\) \(3.9082\) \(\Gamma_0(N)\)-optimal
210210.dk3 210210cd2 \([1, 1, 1, -2164504980, 37641655784277]\) \(9532597152396244075685450929/313550122650789880627200\) \(36888858379742778665909452800\) \([2]\) \(232243200\) \(4.2548\)  
210210.dk2 210210cd3 \([1, 1, 1, -10869288340, 436233243186005]\) \(-1207087636168285491836819264689/236446260657750000000000\) \(-27817666120123629750000000000\) \([2]\) \(348364800\) \(4.4575\)  
210210.dk1 210210cd4 \([1, 1, 1, -173916788340, 27916389331186005]\) \(4944928228995290413834018379264689/189679641808585500000\) \(22315620179138275489500000\) \([2]\) \(696729600\) \(4.8041\)