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Properties

Label 232050.hf
Number of curves $4$
Conductor $232050$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("hf1")
 
E.isogeny_class()
 

Elliptic curves in class 232050.hf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
232050.hf1 232050hf3 \([1, 0, 0, -12910463, 17339064417]\) \(15231025329261085948969/501037266310733880\) \(7828707286105216875000\) \([2]\) \(16515072\) \(2.9743\)  
232050.hf2 232050hf2 \([1, 0, 0, -1975463, -692750583]\) \(54564527576482291369/18314631132033600\) \(286166111438025000000\) \([2, 2]\) \(8257536\) \(2.6277\)  
232050.hf3 232050hf1 \([1, 0, 0, -1775463, -910550583]\) \(39613077168432499369/8661219840000\) \(135331560000000000\) \([2]\) \(4128768\) \(2.2812\) \(\Gamma_0(N)\)-optimal
232050.hf4 232050hf4 \([1, 0, 0, 5759537, -4784565583]\) \(1352279296967264534231/1415615917112986680\) \(-22118998704890416875000\) \([2]\) \(16515072\) \(2.9743\)  

Rank

sage: E.rank()
 

The elliptic curves in class 232050.hf have rank \(1\).

Complex multiplication

The elliptic curves in class 232050.hf do not have complex multiplication.

Modular form 232050.2.a.hf

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{6} + q^{7} + q^{8} + q^{9} + q^{12} - q^{13} + q^{14} + q^{16} + q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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.