Properties

Label 177600bp
Number of curves $6$
Conductor $177600$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bp1")
 
E.isogeny_class()
 

Elliptic curves in class 177600bp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
177600.hd5 177600bp1 \([0, 1, 0, -1352033, -2083475937]\) \(-66730743078481/419010969600\) \(-1716268931481600000000\) \([2]\) \(7962624\) \(2.7581\) \(\Gamma_0(N)\)-optimal
177600.hd4 177600bp2 \([0, 1, 0, -34120033, -76565139937]\) \(1072487167529950801/2554882560000\) \(10464798965760000000000\) \([2, 2]\) \(15925248\) \(3.1047\)  
177600.hd3 177600bp3 \([0, 1, 0, -46920033, -13909139937]\) \(2788936974993502801/1593609593601600\) \(6527424895392153600000000\) \([2, 2]\) \(31850496\) \(3.4513\)  
177600.hd1 177600bp4 \([0, 1, 0, -545608033, -4905523347937]\) \(4385367890843575421521/24975000000\) \(102297600000000000000\) \([2]\) \(31850496\) \(3.4513\)  
177600.hd2 177600bp5 \([0, 1, 0, -485000033, 4093090860063]\) \(3080272010107543650001/15465841417699560\) \(63348086446897397760000000\) \([4]\) \(63700992\) \(3.7978\)  
177600.hd6 177600bp6 \([0, 1, 0, 186359967, -110720339937]\) \(174751791402194852399/102423900876336360\) \(-419528297989473730560000000\) \([2]\) \(63700992\) \(3.7978\)  

Rank

sage: E.rank()
 

The elliptic curves in class 177600bp have rank \(0\).

Complex multiplication

The elliptic curves in class 177600bp do not have complex multiplication.

Modular form 177600.2.a.bp

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{9} + 4 q^{11} - 2 q^{13} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.