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Properties

Label 32856c
Number of curves $6$
Conductor $32856$
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 32856c have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(37\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 - 4 T + 17 T^{2}\) 1.17.ae
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 32856c do not have complex multiplication.

Modular form 32856.2.a.c

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + 2 q^{5} + q^{9} + 4 q^{11} + 2 q^{13} - 2 q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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 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 32856c

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
32856.g5 32856c1 \([0, -1, 0, 913, 12828]\) \(2048/3\) \(-123154867632\) \([2]\) \(25344\) \(0.81353\) \(\Gamma_0(N)\)-optimal
32856.g4 32856c2 \([0, -1, 0, -5932, 133300]\) \(35152/9\) \(5911433646336\) \([2, 2]\) \(50688\) \(1.1601\)  
32856.g3 32856c3 \([0, -1, 0, -33312, -2221380]\) \(1556068/81\) \(212811611268096\) \([2, 2]\) \(101376\) \(1.5067\)  
32856.g2 32856c4 \([0, -1, 0, -88072, 10088668]\) \(28756228/3\) \(7881911528448\) \([2]\) \(101376\) \(1.5067\)  
32856.g6 32856c5 \([0, -1, 0, 21448, -8858292]\) \(207646/6561\) \(-34475481025431552\) \([2]\) \(202752\) \(1.8533\)  
32856.g1 32856c6 \([0, -1, 0, -526152, -146722068]\) \(3065617154/9\) \(47291469170688\) \([2]\) \(202752\) \(1.8533\)