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Properties

Label 74970by
Number of curves $6$
Conductor $74970$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("by1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 74970by have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(5\)\(1 - T\)
\(7\)\(1\)
\(17\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 - 6 T + 11 T^{2}\) 1.11.ag
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 - 2 T + 23 T^{2}\) 1.23.ac
\(29\) \( 1 - 4 T + 29 T^{2}\) 1.29.ae
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 74970by do not have complex multiplication.

Modular form 74970.2.a.by

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} - 4 q^{11} - 6 q^{13} + q^{16} + q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 74970by

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
74970.bd5 74970by1 \([1, -1, 0, 1349451, 585718965]\) \(3168685387909439/3563732336640\) \(-305647498795878973440\) \([2]\) \(3538944\) \(2.6172\) \(\Gamma_0(N)\)-optimal
74970.bd4 74970by2 \([1, -1, 0, -7682229, 5522435253]\) \(584614687782041281/184812061593600\) \(15850613636896150425600\) \([2, 2]\) \(7077888\) \(2.9638\)  
74970.bd3 74970by3 \([1, -1, 0, -48465909, -125662349835]\) \(146796951366228945601/5397929064360000\) \(462959437283316547560000\) \([2, 2]\) \(14155776\) \(3.3103\)  
74970.bd2 74970by4 \([1, -1, 0, -111405429, 452548682613]\) \(1782900110862842086081/328139630024640\) \(28143263213588507221440\) \([2]\) \(14155776\) \(3.3103\)  
74970.bd6 74970by5 \([1, -1, 0, 19007091, -448196784435]\) \(8854313460877886399/1016927675429790600\) \(-87217942059160147604262600\) \([2]\) \(28311552\) \(3.6569\)  
74970.bd1 74970by6 \([1, -1, 0, -768477789, -8199443565027]\) \(585196747116290735872321/836876053125000\) \(71775612834321178125000\) \([2]\) \(28311552\) \(3.6569\)