Properties

Label 303240ce
Number of curves $6$
Conductor $303240$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ce1")
 
E.isogeny_class()
 

Elliptic curves in class 303240ce

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
303240.ce4 303240ce1 \([0, 1, 0, -265455, -52730802]\) \(2748251600896/2205\) \(1659778681680\) \([2]\) \(1769472\) \(1.6502\) \(\Gamma_0(N)\)-optimal
303240.ce3 303240ce2 \([0, 1, 0, -267260, -51979200]\) \(175293437776/4862025\) \(58556991889670400\) \([2, 2]\) \(3538944\) \(1.9968\)  
303240.ce2 303240ce3 \([0, 1, 0, -621040, 114155888]\) \(549871953124/200930625\) \(9679829271557760000\) \([2, 2]\) \(7077888\) \(2.3434\)  
303240.ce5 303240ce4 \([0, 1, 0, 57640, -169982880]\) \(439608956/259416045\) \(-12497363335742100480\) \([2]\) \(7077888\) \(2.3434\)  
303240.ce1 303240ce5 \([0, 1, 0, -8808520, 10057031600]\) \(784478485879202/221484375\) \(21340011621600000000\) \([2]\) \(14155776\) \(2.6900\)  
303240.ce6 303240ce6 \([0, 1, 0, 1905960, 809586288]\) \(7947184069438/7533176175\) \(-725821255436462438400\) \([2]\) \(14155776\) \(2.6900\)  

Rank

sage: E.rank()
 

The elliptic curves in class 303240ce have rank \(1\).

Complex multiplication

The elliptic curves in class 303240ce do not have complex multiplication.

Modular form 303240.2.a.ce

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{5} + q^{7} + q^{9} + 4 q^{11} + 2 q^{13} + q^{15} - 6 q^{17} + O(q^{20})\) 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.