Properties

Label 163170b
Number of curves $4$
Conductor $163170$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 163170b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
163170.ez3 163170b1 \([1, -1, 1, -137822, 24245421]\) \(-3375675045001/999000000\) \(-85680354879000000\) \([2]\) \(2322432\) \(1.9635\) \(\Gamma_0(N)\)-optimal
163170.ez2 163170b2 \([1, -1, 1, -2342822, 1380761421]\) \(16581570075765001/998001000\) \(85594674524121000\) \([2]\) \(4644864\) \(2.3101\)  
163170.ez4 163170b3 \([1, -1, 1, 1019803, -201722979]\) \(1367594037332999/995878502400\) \(-85412636138137190400\) \([2]\) \(6967296\) \(2.5128\)  
163170.ez1 163170b4 \([1, -1, 1, -4624997, -1707755619]\) \(127568139540190201/59114336463360\) \(5070007333951245826560\) \([2]\) \(13934592\) \(2.8594\)  

Rank

sage: E.rank()
 

The elliptic curves in class 163170b have rank \(2\).

Complex multiplication

The elliptic curves in class 163170b do not have complex multiplication.

Modular form 163170.2.a.b

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

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

The vertices are labelled with Cremona labels.