Properties

Label 6440.h
Number of curves $4$
Conductor $6440$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("h1")
 
E.isogeny_class()
 

Elliptic curves in class 6440.h

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6440.h1 6440e4 \([0, 0, 0, -3443, 77758]\) \(4407931365156/100625\) \(103040000\) \([2]\) \(2560\) \(0.65107\)  
6440.h2 6440e3 \([0, 0, 0, -923, -9658]\) \(84923690436/9794435\) \(10029501440\) \([2]\) \(2560\) \(0.65107\)  
6440.h3 6440e2 \([0, 0, 0, -223, 1122]\) \(4790692944/648025\) \(165894400\) \([2, 2]\) \(1280\) \(0.30450\)  
6440.h4 6440e1 \([0, 0, 0, 22, 93]\) \(73598976/276115\) \(-4417840\) \([2]\) \(640\) \(-0.042077\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 6440.h have rank \(1\).

Complex multiplication

The elliptic curves in class 6440.h do not have complex multiplication.

Modular form 6440.2.a.h

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