↑

Properties

Label 7056bp
Number of curves $6$
Conductor $7056$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("bp1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 7056bp have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 7056bp do not have complex multiplication.

Modular form 7056.2.a.bp

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 7056bp

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7056.bd5 7056bp1 \([0, 0, 0, -3675, -173558]\) \(-15625/28\) \(-9836344885248\) \([2]\) \(9216\) \(1.1833\) \(\Gamma_0(N)\)-optimal
7056.bd4 7056bp2 \([0, 0, 0, -74235, -7779926]\) \(128787625/98\) \(34427207098368\) \([2]\) \(18432\) \(1.5299\)  
7056.bd6 7056bp3 \([0, 0, 0, 31605, 3629626]\) \(9938375/21952\) \(-7711694390034432\) \([2]\) \(27648\) \(1.7326\)  
7056.bd3 7056bp4 \([0, 0, 0, -250635, 39587002]\) \(4956477625/941192\) \(330638896972726272\) \([2]\) \(55296\) \(2.0792\)  
7056.bd2 7056bp5 \([0, 0, 0, -1203195, 509453098]\) \(-548347731625/1835008\) \(-644634698399612928\) \([2]\) \(82944\) \(2.2819\)  
7056.bd1 7056bp6 \([0, 0, 0, -19266555, 32550241066]\) \(2251439055699625/25088\) \(8813365017182208\) \([2]\) \(165888\) \(2.6285\)