Properties

Label 34200.bp
Number of curves $4$
Conductor $34200$
CM no
Rank $1$
Graph

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

Elliptic curves in class 34200.bp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
34200.bp1 34200z4 \([0, 0, 0, -6840075, -6885562250]\) \(3034301922374404/1425\) \(16621200000000\) \([2]\) \(393216\) \(2.3105\)  
34200.bp2 34200z3 \([0, 0, 0, -513075, -61465250]\) \(1280615525284/601171875\) \(7012068750000000000\) \([2]\) \(393216\) \(2.3105\)  
34200.bp3 34200z2 \([0, 0, 0, -427575, -107549750]\) \(2964647793616/2030625\) \(5921302500000000\) \([2, 2]\) \(196608\) \(1.9639\)  
34200.bp4 34200z1 \([0, 0, 0, -21450, -2363375]\) \(-5988775936/9774075\) \(-1781325168750000\) \([2]\) \(98304\) \(1.6173\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 34200.bp have rank \(1\).

Complex multiplication

The elliptic curves in class 34200.bp do not have complex multiplication.

Modular form 34200.2.a.bp

sage: E.q_eigenform(10)
 
\(q + 2 q^{13} - 6 q^{17} + q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.