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Properties

Label 90.c
Number of curves $8$
Conductor $90$
CM no
Rank $0$
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 90.c have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 90.c do not have complex multiplication.

Modular form 90.2.a.c

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} - 4 q^{7} + q^{8} + q^{10} + 2 q^{13} - 4 q^{14} + q^{16} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 90.c

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
90.c1 90c8 \([1, -1, 1, -48002, 4059929]\) \(16778985534208729/81000\) \(59049000\) \([6]\) \(192\) \(1.1135\)  
90.c2 90c7 \([1, -1, 1, -4082, 14681]\) \(10316097499609/5859375000\) \(4271484375000\) \([6]\) \(192\) \(1.1135\)  
90.c3 90c6 \([1, -1, 1, -3002, 63929]\) \(4102915888729/9000000\) \(6561000000\) \([2, 6]\) \(96\) \(0.76690\)  
90.c4 90c4 \([1, -1, 1, -2597, -50281]\) \(2656166199049/33750\) \(24603750\) \([2]\) \(64\) \(0.56417\)  
90.c5 90c5 \([1, -1, 1, -617, 5231]\) \(35578826569/5314410\) \(3874204890\) \([2]\) \(64\) \(0.56417\)  
90.c6 90c2 \([1, -1, 1, -167, -709]\) \(702595369/72900\) \(53144100\) \([2, 2]\) \(32\) \(0.21759\)  
90.c7 90c3 \([1, -1, 1, -122, 1721]\) \(-273359449/1536000\) \(-1119744000\) \([12]\) \(48\) \(0.42032\)  
90.c8 90c1 \([1, -1, 1, 13, -61]\) \(357911/2160\) \(-1574640\) \([4]\) \(16\) \(-0.12898\) \(\Gamma_0(N)\)-optimal