Properties

Label 816b
Number of curves $4$
Conductor $816$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 816b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
816.f3 816b1 \([0, -1, 0, -52, -128]\) \(61918288/153\) \(39168\) \([2]\) \(128\) \(-0.24084\) \(\Gamma_0(N)\)-optimal
816.f2 816b2 \([0, -1, 0, -72, 0]\) \(40873252/23409\) \(23970816\) \([2, 2]\) \(256\) \(0.10573\)  
816.f1 816b3 \([0, -1, 0, -752, 8160]\) \(22994537186/111537\) \(228427776\) \([2]\) \(512\) \(0.45230\)  
816.f4 816b4 \([0, -1, 0, 288, -288]\) \(1285471294/751689\) \(-1539459072\) \([4]\) \(512\) \(0.45230\)  

Rank

sage: E.rank()
 

The elliptic curves in class 816b have rank \(0\).

Complex multiplication

The elliptic curves in class 816b do not have complex multiplication.

Modular form 816.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} + 4 q^{7} + q^{9} - 4 q^{11} + 6 q^{13} - 2 q^{15} + q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.