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Properties

Label 210a
Number of curves $8$
Conductor $210$
CM no
Rank $0$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -41, -39]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -41, -39]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -41, -39]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 210a have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1 + T\)
\(7\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 210a do not have complex multiplication.

Modular form 210.2.a.a

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} + q^{12} + 2 q^{13} + q^{14} - q^{15} + q^{16} - 6 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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 comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 210a

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
210.d7 210a1 \([1, 0, 0, -41, -39]\) \(7633736209/3870720\) \(3870720\) \([6]\) \(48\) \(-0.043427\) \(\Gamma_0(N)\)-optimal
210.d5 210a2 \([1, 0, 0, -361, 2585]\) \(5203798902289/57153600\) \(57153600\) \([2, 6]\) \(96\) \(0.30315\)  
210.d4 210a3 \([1, 0, 0, -2681, -53655]\) \(2131200347946769/2058000\) \(2058000\) \([2]\) \(144\) \(0.50588\)  
210.d2 210a4 \([1, 0, 0, -5761, 167825]\) \(21145699168383889/2593080\) \(2593080\) \([6]\) \(192\) \(0.64972\)  
210.d6 210a5 \([1, 0, 0, -81, 6561]\) \(-58818484369/18600435000\) \(-18600435000\) \([6]\) \(192\) \(0.64972\)  
210.d3 210a6 \([1, 0, 0, -2701, -52819]\) \(2179252305146449/66177562500\) \(66177562500\) \([2, 2]\) \(288\) \(0.85245\)  
210.d1 210a7 \([1, 0, 0, -6451, 124931]\) \(29689921233686449/10380965400750\) \(10380965400750\) \([2]\) \(576\) \(1.1990\)  
210.d8 210a8 \([1, 0, 0, 729, -176985]\) \(42841933504271/13565917968750\) \(-13565917968750\) \([2]\) \(576\) \(1.1990\)