Properties

Label 4400.e
Number of curves $4$
Conductor $4400$
CM no
Rank $1$
Graph

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

Elliptic curves in class 4400.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4400.e1 4400t4 \([0, 1, 0, -177508, 28726488]\) \(154639330142416/33275\) \(133100000000\) \([2]\) \(20736\) \(1.5192\)  
4400.e2 4400t3 \([0, 1, 0, -11133, 442738]\) \(610462990336/8857805\) \(2214451250000\) \([2]\) \(10368\) \(1.1726\)  
4400.e3 4400t2 \([0, 1, 0, -2508, 26488]\) \(436334416/171875\) \(687500000000\) \([2]\) \(6912\) \(0.96986\)  
4400.e4 4400t1 \([0, 1, 0, -1133, -14762]\) \(643956736/15125\) \(3781250000\) \([2]\) \(3456\) \(0.62329\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 4400.e have rank \(1\).

Complex multiplication

The elliptic curves in class 4400.e do not have complex multiplication.

Modular form 4400.2.a.e

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.