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Properties

Label 210210.fp
Number of curves $2$
Conductor $210210$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("fp1")
 
E.isogeny_class()
 

Elliptic curves in class 210210.fp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
210210.fp1 210210h2 \([1, 0, 0, -790504310, 8554628032572]\) \(464352938845529653759213009/2445173327025000\) \(287672196751164225000\) \([2]\) \(58060800\) \(3.5424\)  
210210.fp2 210210h1 \([1, 0, 0, -49379310, 133817557572]\) \(-113180217375258301213009/260161419375000000\) \(-30607730828049375000000\) \([2]\) \(29030400\) \(3.1959\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 210210.fp have rank \(1\).

Complex multiplication

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

Modular form 210210.2.a.fp

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

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.