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Properties

Label 324870e
Number of curves $4$
Conductor $324870$
CM no
Rank $1$
Graph

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

Elliptic curves in class 324870e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
324870.e3 324870e1 \([1, 1, 0, -1528923, -728166963]\) \(3359640110447736361/679479899280\) \(79940130670392720\) \([2]\) \(7667712\) \(2.2412\) \(\Gamma_0(N)\)-optimal
324870.e2 324870e2 \([1, 1, 0, -1694543, -560923887]\) \(4573973602374520681/1494559259820900\) \(175833402358669064100\) \([2, 2]\) \(15335424\) \(2.5878\)  
324870.e1 324870e3 \([1, 1, 0, -10899193, 13424621323]\) \(1217078423760736099081/42121389971467170\) \(4955539408753141083330\) \([2]\) \(30670848\) \(2.9344\)  
324870.e4 324870e4 \([1, 1, 0, 4860187, -3839599833]\) \(107918095079651282999/116643384110321250\) \(-13722977497195184741250\) \([2]\) \(30670848\) \(2.9344\)  

Rank

sage: E.rank()
 

The elliptic curves in class 324870e have rank \(1\).

Complex multiplication

The elliptic curves in class 324870e do not have complex multiplication.

Modular form 324870.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} - 4 q^{11} - q^{12} + q^{13} + q^{15} + q^{16} - q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.