Properties

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

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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 210.d

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

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 210.2.a.d

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{7} + q^{8} + q^{9} - q^{10} + q^{12} + 2 q^{13} + q^{14} - q^{15} + q^{16} - 6 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.