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Properties

Label 318402.bc
Number of curves $6$
Conductor $318402$
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 318402.bc have rank \(0\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(7\)\(1\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 + 6 T + 11 T^{2}\) 1.11.g
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 318402.bc do not have complex multiplication.

Modular form 318402.2.a.bc

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{8} - 6 q^{11} - 4 q^{13} + q^{16} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 318402.bc

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
318402.bc1 318402bc6 \([1, -1, 0, -1558054101627, 748552285208935269]\) \(103665426767620308239307625/5961940992\) \(24056090417034333924360192\) \([2]\) \(2866544640\) \(5.2567\)  
318402.bc2 318402bc5 \([1, -1, 0, -97378558587, 11696103010564965]\) \(25309080274342544331625/191933498523648\) \(774440673052304193436356968448\) \([2]\) \(1433272320\) \(4.9101\)  
318402.bc3 318402bc4 \([1, -1, 0, -19253055852, 1024827113040912]\) \(195607431345044517625/752875610010048\) \(3037809963480697619886261094848\) \([2]\) \(955514880\) \(4.7074\)  
318402.bc4 318402bc3 \([1, -1, 0, -1779154092, -887425490736]\) \(154357248921765625/89242711068672\) \(360089227553569919768799055872\) \([2]\) \(477757440\) \(4.3608\)  
318402.bc5 318402bc2 \([1, -1, 0, -1264138857, -16207877571327]\) \(55369510069623625/3916046302812\) \(15801022530103311460308354012\) \([2]\) \(318504960\) \(4.1581\)  
318402.bc6 318402bc1 \([1, -1, 0, -1241850717, -16843878561483]\) \(52492168638015625/293197968\) \(1183037007203364701274768\) \([2]\) \(159252480\) \(3.8115\) \(\Gamma_0(N)\)-optimal