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Properties

Label 61347.i
Number of curves $6$
Conductor $61347$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("i1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 61347.i have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(3\)\(1 - T\)
\(11\)\(1\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + T + 2 T^{2}\) 1.2.b
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 61347.i do not have complex multiplication.

Modular form 61347.2.a.i

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} + 2 q^{5} - q^{6} + 3 q^{8} + q^{9} - 2 q^{10} - q^{12} + 2 q^{15} - q^{16} + 6 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 61347.i

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
61347.i1 61347z4 \([1, 0, 0, -140362362, -640076749443]\) \(35765103905346817/1287\) \(11005119726978663\) \([2]\) \(5160960\) \(3.0225\)  
61347.i2 61347z6 \([1, 0, 0, -61531467, 179889890478]\) \(3013001140430737/108679952667\) \(929320816645461583340283\) \([2]\) \(10321920\) \(3.3691\)  
61347.i3 61347z3 \([1, 0, 0, -9693252, -7774815465]\) \(11779205551777/3763454409\) \(32181248141469095195241\) \([2, 2]\) \(5160960\) \(3.0225\)  
61347.i4 61347z2 \([1, 0, 0, -8773047, -10000791360]\) \(8732907467857/1656369\) \(14163589088621539281\) \([2, 2]\) \(2580480\) \(2.6759\)  
61347.i5 61347z1 \([1, 0, 0, -491202, -190117773]\) \(-1532808577/938223\) \(-8022732280967445327\) \([4]\) \(1290240\) \(2.3294\) \(\Gamma_0(N)\)-optimal
61347.i6 61347z5 \([1, 0, 0, 27421683, -52973383308]\) \(266679605718863/296110251723\) \(-2532034788342971977606827\) \([2]\) \(10321920\) \(3.3691\)