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Properties

Label 215985.bh
Number of curves $4$
Conductor $215985$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bh1")
 
E.isogeny_class()
 

Elliptic curves in class 215985.bh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
215985.bh1 215985cc3 \([1, 1, 0, -7405323, 7753385358]\) \(25351269426118370449/27551475\) \(48809118602475\) \([2]\) \(4423680\) \(2.3453\)  
215985.bh2 215985cc4 \([1, 1, 0, -577293, 56470572]\) \(12010404962647729/6166198828125\) \(10923797362151953125\) \([2]\) \(4423680\) \(2.3453\)  
215985.bh3 215985cc2 \([1, 1, 0, -462948, 120938283]\) \(6193921595708449/6452105625\) \(11430298693130625\) \([2, 2]\) \(2211840\) \(1.9988\)  
215985.bh4 215985cc1 \([1, 1, 0, -21903, 2826432]\) \(-656008386769/1581036975\) \(-2800903444467975\) \([2]\) \(1105920\) \(1.6522\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 215985.bh have rank \(1\).

Complex multiplication

The elliptic curves in class 215985.bh do not have complex multiplication.

Modular form 215985.2.a.bh

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.