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Properties

Label 7800.r
Number of curves $6$
Conductor $7800$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("r1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 7800.r have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1\)
\(13\)\(1 + T\)
 
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
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 7800.r do not have complex multiplication.

Modular form 7800.2.a.r

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} + q^{9} - 4 q^{11} - q^{13} - 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 7800.r

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7800.r1 7800e4 \([0, 1, 0, -60840008, 182634907488]\) \(1556580279686303289604/114075\) \(1825200000000\) \([2]\) \(368640\) \(2.7223\)  
7800.r2 7800e5 \([0, 1, 0, -13364008, -15704090512]\) \(8248670337458940482/1446075439453125\) \(46274414062500000000000\) \([2]\) \(737280\) \(3.0689\)  
7800.r3 7800e3 \([0, 1, 0, -3887008, 2719197488]\) \(405929061432816484/35083409765625\) \(561334556250000000000\) \([2, 2]\) \(368640\) \(2.7223\)  
7800.r4 7800e2 \([0, 1, 0, -3802508, 2852707488]\) \(1520107298839022416/13013105625\) \(52052422500000000\) \([2, 2]\) \(184320\) \(2.3757\)  
7800.r5 7800e1 \([0, 1, 0, -232383, 46589238]\) \(-5551350318708736/550618236675\) \(-137654559168750000\) \([2]\) \(92160\) \(2.0291\) \(\Gamma_0(N)\)-optimal
7800.r6 7800e6 \([0, 1, 0, 4237992, 12599197488]\) \(263059523447441758/2294739983908125\) \(-73431679485060000000000\) \([2]\) \(737280\) \(3.0689\)