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Properties

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

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Show commands: SageMath
E = EllipticCurve("a1")
 
E.isogeny_class()
 

Elliptic curves in class 210210.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
210210.a1 210210fn4 \([1, 1, 0, -111307053, 451947074013]\) \(1296294060988412126189641/647824320\) \(76215883423680\) \([2]\) \(14929920\) \(2.8998\)  
210210.a2 210210fn3 \([1, 1, 0, -6956653, 7059578653]\) \(-316472948332146183241/7074906009600\) \(-832355617123430400\) \([2]\) \(7464960\) \(2.5533\)  
210210.a3 210210fn2 \([1, 1, 0, -1376778, 616964328]\) \(2453170411237305241/19353090685500\) \(2276871766058389500\) \([2]\) \(4976640\) \(2.3505\)  
210210.a4 210210fn1 \([1, 1, 0, -29278, 22177828]\) \(-23592983745241/1794399750000\) \(-211109336187750000\) \([2]\) \(2488320\) \(2.0039\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 210210.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} - q^{11} - q^{12} - q^{13} + q^{15} + q^{16} - 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

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 LMFDB 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

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

The vertices are labelled with LMFDB labels.