Properties

Label 291312.bg
Number of curves $4$
Conductor $291312$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bg1")
 
E.isogeny_class()
 

Elliptic curves in class 291312.bg

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
291312.bg1 291312bg3 \([0, 0, 0, -471736670691, 124708934902298434]\) \(322159999717985454060440834/4250799\) \(153186882740639741952\) \([2]\) \(637009920\) \(4.8569\)  
291312.bg2 291312bg4 \([0, 0, 0, -29559387891, 1938047760363346]\) \(79260902459030376659234/842751810121431609\) \(30370413354414930637009923704832\) \([2]\) \(637009920\) \(4.8569\)  
291312.bg3 291312bg2 \([0, 0, 0, -29483542731, 1948576995038410]\) \(157304700372188331121828/18069292138401\) \(325583323983514337224909824\) \([2, 2]\) \(318504960\) \(4.5103\)  
291312.bg4 291312bg1 \([0, 0, 0, -1837981911, 30610922029270]\) \(-152435594466395827792/1646846627220711\) \(-7418467127470538624984706816\) \([2]\) \(159252480\) \(4.1638\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 291312.bg have rank \(0\).

Complex multiplication

The elliptic curves in class 291312.bg do not have complex multiplication.

Modular form 291312.2.a.bg

sage: E.q_eigenform(10)
 
\(q - 2 q^{5} - q^{7} + 2 q^{13} + O(q^{20})\) 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.