Properties

Label 8470.ba
Number of curves $4$
Conductor $8470$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("ba1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 8470.ba

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
8470.ba1 8470be4 \([1, -1, 1, -395266972, 3024809938571]\) \(3855131356812007128171561/8967612500\) \(15886672568112500\) \([2]\) \(1228800\) \(3.2390\)  
8470.ba2 8470be3 \([1, -1, 1, -26008852, 41998530379]\) \(1098325674097093229481/205612182617187500\) \(364254523849487304687500\) \([2]\) \(1228800\) \(3.2390\)  
8470.ba3 8470be2 \([1, -1, 1, -24704472, 47266138571]\) \(941226862950447171561/45393906250000\) \(80418073950156250000\) \([2, 2]\) \(614400\) \(2.8924\)  
8470.ba4 8470be1 \([1, -1, 1, -1462792, 819965259]\) \(-195395722614328041/50730248800000\) \(-89871730294376800000\) \([4]\) \(307200\) \(2.5458\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 8470.ba have rank \(1\).

Complex multiplication

The elliptic curves in class 8470.ba do not have complex multiplication.

Modular form 8470.2.a.ba

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} - 3q^{9} + q^{10} - 2q^{13} + q^{14} + q^{16} - 6q^{17} - 3q^{18} - 4q^{19} + O(q^{20})\)  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.