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Properties

Label 177870.bb
Number of curves $4$
Conductor $177870$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("bb1")
 
E.isogeny_class()
 

Elliptic curves in class 177870.bb

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
177870.bb1 177870hb3 \([1, 1, 0, -574111122, 5294461718484]\) \(100407751863770656369/166028940000\) \(34604146838453775660000\) \([2]\) \(58982400\) \(3.5876\)  
177870.bb2 177870hb2 \([1, 1, 0, -36232242, 81016886196]\) \(25238585142450289/995844326400\) \(207556244706414977049600\) \([2, 2]\) \(29491200\) \(3.2410\)  
177870.bb3 177870hb1 \([1, 1, 0, -5875762, -3780905036]\) \(107639597521009/32699842560\) \(6815379014890778787840\) \([2]\) \(14745600\) \(2.8945\) \(\Gamma_0(N)\)-optimal
177870.bb4 177870hb4 \([1, 1, 0, 15942958, 295237822356]\) \(2150235484224911/181905111732960\) \(-37913096337739002589033440\) \([2]\) \(58982400\) \(3.5876\)  

Rank

sage: E.rank()
 

The elliptic curves in class 177870.bb have rank \(2\).

Complex multiplication

The elliptic curves in class 177870.bb do not have complex multiplication.

Modular form 177870.2.a.bb

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.