Properties

Label 167310.fr
Number of curves $8$
Conductor $167310$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("fr1")
 
E.isogeny_class()
 

Elliptic curves in class 167310.fr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
167310.fr1 167310w8 \([1, -1, 1, -1592487032, 13752640434719]\) \(126929854754212758768001/50235797102795981820\) \(176766897634325236684994425020\) \([2]\) \(222953472\) \(4.3104\)  
167310.fr2 167310w6 \([1, -1, 1, -1390041932, 19941630075839]\) \(84415028961834287121601/30783551683856400\) \(108319430428990751919920400\) \([2, 2]\) \(111476736\) \(3.9639\)  
167310.fr3 167310w3 \([1, -1, 1, -1389920252, 19945296829631]\) \(84392862605474684114881/11228954880\) \(39511814926550503680\) \([4]\) \(55738368\) \(3.6173\)  
167310.fr4 167310w7 \([1, -1, 1, -1189543712, 25895946014111]\) \(-52902632853833942200321/51713453577420277500\) \(-181966392135310731928013677500\) \([2]\) \(222953472\) \(4.3104\)  
167310.fr5 167310w5 \([1, -1, 1, -717759932, -7400537597761]\) \(11621808143080380273601/1335706803288000\) \(4700009980594904286168000\) \([2]\) \(74317824\) \(3.7611\)  
167310.fr6 167310w2 \([1, -1, 1, -48519932, -95649149761]\) \(3590017885052913601/954068544000000\) \(3357122736766353984000000\) \([2, 2]\) \(37158912\) \(3.4146\)  
167310.fr7 167310w1 \([1, -1, 1, -17369852, 26658524351]\) \(164711681450297281/8097103872000\) \(28491633731768942592000\) \([4]\) \(18579456\) \(3.0680\) \(\Gamma_0(N)\)-optimal
167310.fr8 167310w4 \([1, -1, 1, 122318788, -618825645889]\) \(57519563401957999679/80296734375000000\) \(-282543633110705484375000000\) \([2]\) \(74317824\) \(3.7611\)  

Rank

sage: E.rank()
 

The elliptic curves in class 167310.fr have rank \(0\).

Complex multiplication

The elliptic curves in class 167310.fr do not have complex multiplication.

Modular form 167310.2.a.fr

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} + 4 q^{7} + q^{8} + q^{10} - q^{11} + 4 q^{14} + q^{16} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 3 & 6 & 12 & 12 \\ 2 & 1 & 2 & 2 & 6 & 3 & 6 & 6 \\ 4 & 2 & 1 & 4 & 12 & 6 & 3 & 12 \\ 4 & 2 & 4 & 1 & 12 & 6 & 12 & 3 \\ 3 & 6 & 12 & 12 & 1 & 2 & 4 & 4 \\ 6 & 3 & 6 & 6 & 2 & 1 & 2 & 2 \\ 12 & 6 & 3 & 12 & 4 & 2 & 1 & 4 \\ 12 & 6 & 12 & 3 & 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.