Properties

Label 448.a
Number of curves $6$
Conductor $448$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 448.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
448.a1 448f6 \([0, 1, 0, -174753, 28059871]\) \(2251439055699625/25088\) \(6576668672\) \([2]\) \(1152\) \(1.4528\)  
448.a2 448f5 \([0, 1, 0, -10913, 436447]\) \(-548347731625/1835008\) \(-481036337152\) \([2]\) \(576\) \(1.1062\)  
448.a3 448f4 \([0, 1, 0, -2273, 33439]\) \(4956477625/941192\) \(246727835648\) \([2]\) \(384\) \(0.90352\)  
448.a4 448f2 \([0, 1, 0, -673, -6945]\) \(128787625/98\) \(25690112\) \([2]\) \(128\) \(0.35421\)  
448.a5 448f1 \([0, 1, 0, -33, -161]\) \(-15625/28\) \(-7340032\) \([2]\) \(64\) \(0.0076359\) \(\Gamma_0(N)\)-optimal
448.a6 448f3 \([0, 1, 0, 287, 3231]\) \(9938375/21952\) \(-5754585088\) \([2]\) \(192\) \(0.55694\)  

Rank

sage: E.rank()
 

The elliptic curves in class 448.a have rank \(0\).

Complex multiplication

The elliptic curves in class 448.a do not have complex multiplication.

Modular form 448.2.a.a

sage: E.q_eigenform(10)
 
\(q - 2q^{3} - q^{7} + q^{9} + 4q^{13} + 6q^{17} + 2q^{19} + O(q^{20})\)  Toggle raw display

Isogeny matrix

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.