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Properties

Label 83205.h
Number of curves $2$
Conductor $83205$
CM no
Rank $0$
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Show commands: SageMath
E = EllipticCurve("h1")
 
E.isogeny_class()
 

Elliptic curves in class 83205.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
83205.h1 83205c2 \([1, -1, 1, -1376003, -618774524]\) \(2315685267/9245\) \(1150294230320102415\) \([2]\) \(1419264\) \(2.3223\)  
83205.h2 83205c1 \([1, -1, 1, -127928, 769906]\) \(1860867/1075\) \(133755143060477025\) \([2]\) \(709632\) \(1.9757\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 83205.h have rank \(0\).

Complex multiplication

The elliptic curves in class 83205.h do not have complex multiplication.

Modular form 83205.2.a.h

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} - q^{5} + 4 q^{7} + 3 q^{8} + q^{10} + 2 q^{13} - 4 q^{14} - q^{16} + 2 q^{17} - 6 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.