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Properties

Label 317130cu
Number of curves $6$
Conductor $317130$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cu1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 317130cu have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(11\)\(1 + T\)
\(31\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 317130cu do not have complex multiplication.

Modular form 317130.2.a.cu

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + q^{10} - q^{11} + q^{12} - 6 q^{13} + q^{15} + q^{16} - 2 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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 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 317130cu

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
317130.cu6 317130cu1 \([1, 0, 0, 245035, -4416543]\) \(1833318007919/1070530560\) \(-950099812622991360\) \([2]\) \(5898240\) \(2.1390\) \(\Gamma_0(N)\)-optimal
317130.cu5 317130cu2 \([1, 0, 0, -985045, -35660575]\) \(119102750067601/68309049600\) \(60624532965611577600\) \([2, 2]\) \(11796480\) \(2.4856\)  
317130.cu3 317130cu3 \([1, 0, 0, -10287525, 12647340657]\) \(135670761487282321/643043610000\) \(570703570918528410000\) \([2, 2]\) \(23592960\) \(2.8322\)  
317130.cu2 317130cu4 \([1, 0, 0, -11363845, -14713359535]\) \(182864522286982801/463015182960\) \(410927679235888475760\) \([2]\) \(23592960\) \(2.8322\)  
317130.cu1 317130cu5 \([1, 0, 0, -164412705, 811416498525]\) \(553808571467029327441/12529687500\) \(11120143778029687500\) \([2]\) \(47185920\) \(3.1788\)  
317130.cu4 317130cu6 \([1, 0, 0, -5002025, 25627471557]\) \(-15595206456730321/310672490129100\) \(-275722978575012415217100\) \([2]\) \(47185920\) \(3.1788\)