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Properties

Label 720.c
Number of curves $8$
Conductor $720$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("c1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 720.c have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 720.c do not have complex multiplication.

Modular form 720.2.a.c

Copy content sage:E.q_eigenform(10)
 
\(q - q^{5} - 4 q^{11} - 2 q^{13} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 720.c

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
720.c1 720h7 \([0, 0, 0, -311043, -66769598]\) \(1114544804970241/405\) \(1209323520\) \([2]\) \(2048\) \(1.5333\)  
720.c2 720h5 \([0, 0, 0, -19443, -1042958]\) \(272223782641/164025\) \(489776025600\) \([2, 2]\) \(1024\) \(1.1867\)  
720.c3 720h8 \([0, 0, 0, -15843, -1441118]\) \(-147281603041/215233605\) \(-642684100792320\) \([2]\) \(2048\) \(1.5333\)  
720.c4 720h4 \([0, 0, 0, -11523, 476098]\) \(56667352321/15\) \(44789760\) \([2]\) \(512\) \(0.84018\)  
720.c5 720h3 \([0, 0, 0, -1443, -9758]\) \(111284641/50625\) \(151165440000\) \([2, 2]\) \(512\) \(0.84018\)  
720.c6 720h2 \([0, 0, 0, -723, 7378]\) \(13997521/225\) \(671846400\) \([2, 2]\) \(256\) \(0.49360\)  
720.c7 720h1 \([0, 0, 0, -3, 322]\) \(-1/15\) \(-44789760\) \([2]\) \(128\) \(0.14703\) \(\Gamma_0(N)\)-optimal
720.c8 720h6 \([0, 0, 0, 5037, -73262]\) \(4733169839/3515625\) \(-10497600000000\) \([2]\) \(1024\) \(1.1867\)