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Properties

Label 303450.gd
Number of curves $4$
Conductor $303450$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("gd1")
 
E.isogeny_class()
 

Elliptic curves in class 303450.gd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
303450.gd1 303450gd4 \([1, 0, 0, -27005045213, -1708102899953583]\) \(5774905528848578698851241/31070538632700000\) \(11718238595530654785937500000\) \([2]\) \(796262400\) \(4.5804\)  
303450.gd2 303450gd3 \([1, 0, 0, -5431773213, 123214305750417]\) \(46993202771097749198761/9805297851562500000\) \(3698063335275650024414062500000\) \([2]\) \(796262400\) \(4.5804\)  
303450.gd3 303450gd2 \([1, 0, 0, -1717545213, -25700237453583]\) \(1485712211163154851241/103233690000000000\) \(38934536179681406250000000000\) \([2, 2]\) \(398131200\) \(4.2338\)  
303450.gd4 303450gd1 \([1, 0, 0, 95062787, -1739372301583]\) \(251907898698209879/3611226931200000\) \(-1361972487914035200000000000\) \([2]\) \(199065600\) \(3.8873\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 303450.gd have rank \(1\).

Complex multiplication

The elliptic curves in class 303450.gd do not have complex multiplication.

Modular form 303450.2.a.gd

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + q^{8} + q^{9} + 4 q^{11} + q^{12} - 6 q^{13} - q^{14} + q^{16} + q^{18} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.