Properties

Label 283920ej
Number of curves $6$
Conductor $283920$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("ej1")
 
E.isogeny_class()
 

Elliptic curves in class 283920ej

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
283920.ej4 283920ej1 \([0, 1, 0, -29631, -1973100]\) \(37256083456/525\) \(40545195600\) \([2]\) \(589824\) \(1.1757\) \(\Gamma_0(N)\)-optimal
283920.ej3 283920ej2 \([0, 1, 0, -30476, -1855476]\) \(2533446736/275625\) \(340579643040000\) \([2, 2]\) \(1179648\) \(1.5222\)  
283920.ej2 283920ej3 \([0, 1, 0, -114976, 12982724]\) \(34008619684/4862025\) \(24031299612902400\) \([2, 2]\) \(2359296\) \(1.8688\)  
283920.ej5 283920ej4 \([0, 1, 0, 40504, -9152220]\) \(1486779836/8203125\) \(-40545195600000000\) \([2]\) \(2359296\) \(1.8688\)  
283920.ej1 283920ej5 \([0, 1, 0, -1771176, 906668244]\) \(62161150998242/1607445\) \(15890083825674240\) \([2]\) \(4718592\) \(2.2154\)  
283920.ej6 283920ej6 \([0, 1, 0, 189224, 70294004]\) \(75798394558/259416045\) \(-2564406683136829440\) \([2]\) \(4718592\) \(2.2154\)  

Rank

sage: E.rank()
 

The elliptic curves in class 283920ej have rank \(2\).

Complex multiplication

The elliptic curves in class 283920ej do not have complex multiplication.

Modular form 283920.2.a.ej

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{5} - q^{7} + q^{9} - 4 q^{11} - q^{15} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.