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Properties

Label 303450bc
Number of curves $6$
Conductor $303450$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("bc1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 303450bc have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(7\)\(1 - T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 8 T + 23 T^{2}\) 1.23.ai
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 303450bc do not have complex multiplication.

Modular form 303450.2.a.bc

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} + q^{7} - q^{8} + q^{9} - 4 q^{11} - q^{12} + 2 q^{13} - q^{14} + q^{16} - q^{18} + 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 303450bc

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
303450.bc5 303450bc1 \([1, 1, 0, -507513050, 4394954356500]\) \(38331145780597164097/55468445663232\) \(20919897414359580672000000\) \([2]\) \(141557760\) \(3.7607\) \(\Gamma_0(N)\)-optimal
303450.bc4 303450bc2 \([1, 1, 0, -655481050, 1623069812500]\) \(82582985847542515777/44772582831427584\) \(16885957928153104332864000000\) \([2, 2]\) \(283115520\) \(4.1073\)  
303450.bc6 303450bc3 \([1, 1, 0, 2528142950, 12768937436500]\) \(4738217997934888496063/2928751705237796928\) \(-1104577287016327886837313000000\) \([2]\) \(566231040\) \(4.4539\)  
303450.bc2 303450bc4 \([1, 1, 0, -6206593050, -186920449267500]\) \(70108386184777836280897/552468975892674624\) \(208363406655762036427329000000\) \([2, 2]\) \(566231040\) \(4.4539\)  
303450.bc3 303450bc5 \([1, 1, 0, -2114064050, -429734287366500]\) \(-2770540998624539614657/209924951154647363208\) \(-79173093645577037501580646125000\) \([2]\) \(1132462080\) \(4.8005\)  
303450.bc1 303450bc6 \([1, 1, 0, -99116914050, -12010780810048500]\) \(285531136548675601769470657/17941034271597192\) \(6766452385344405657910125000\) \([2]\) \(1132462080\) \(4.8005\)