Properties

Label 400.e
Number of curves $4$
Conductor $400$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 400.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
400.e1 400a3 \([0, 0, 0, -2675, 53250]\) \(132304644/5\) \(80000000\) \([4]\) \(192\) \(0.60250\)  
400.e2 400a2 \([0, 0, 0, -175, 750]\) \(148176/25\) \(100000000\) \([2, 2]\) \(96\) \(0.25593\)  
400.e3 400a1 \([0, 0, 0, -50, -125]\) \(55296/5\) \(1250000\) \([2]\) \(48\) \(-0.090642\) \(\Gamma_0(N)\)-optimal
400.e4 400a4 \([0, 0, 0, 325, 4250]\) \(237276/625\) \(-10000000000\) \([2]\) \(192\) \(0.60250\)  

Rank

sage: E.rank()
 

The elliptic curves in class 400.e have rank \(1\).

Complex multiplication

The elliptic curves in class 400.e do not have complex multiplication.

Modular form 400.2.a.e

sage: E.q_eigenform(10)
 
\(q - 4 q^{7} - 3 q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{17} - 4 q^{19} + O(q^{20})\) Copy content 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.