Properties

Label 327990.b
Number of curves $4$
Conductor $327990$
CM no
Rank $1$
Graph

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

Elliptic curves in class 327990.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
327990.b1 327990b4 \([1, 1, 0, -54430378, -154569469772]\) \(29981943972267024529/4007065140000\) \(2383495794038129940000\) \([2]\) \(32256000\) \(3.1211\)  
327990.b2 327990b3 \([1, 1, 0, -21866858, 37791101172]\) \(1943993954077461649/87266819409120\) \(51908339334040016087520\) \([2]\) \(32256000\) \(3.1211\)  
327990.b3 327990b2 \([1, 1, 0, -3701258, -1966130988]\) \(9427227449071249/2652468249600\) \(1577749973074128921600\) \([2, 2]\) \(16128000\) \(2.7745\)  
327990.b4 327990b1 \([1, 1, 0, 604662, -201564972]\) \(41102915774831/53367275520\) \(-31744100057528401920\) \([2]\) \(8064000\) \(2.4279\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 327990.b have rank \(1\).

Complex multiplication

The elliptic curves in class 327990.b do not have complex multiplication.

Modular form 327990.2.a.b

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.