Properties

Label 333270.ea
Number of curves $4$
Conductor $333270$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("ea1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 333270.ea

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.ea1 333270ea3 \([1, -1, 1, -598188803, 3157747962027]\) \(219353215817909485369/87028564162480920\) \(9391962779991888106110914520\) \([2]\) \(311427072\) \(4.0658\)  
333270.ea2 333270ea2 \([1, -1, 1, -271584203, -1687888525413]\) \(20527812941011798969/474091398849600\) \(51163072896350548591617600\) \([2, 2]\) \(155713536\) \(3.7192\)  
333270.ea3 333270ea1 \([1, -1, 1, -270060683, -1708137325029]\) \(20184279492242626489/11148103680\) \(1203082870982136238080\) \([2]\) \(77856768\) \(3.3727\) \(\Gamma_0(N)\)-optimal
333270.ea4 333270ea4 \([1, -1, 1, 30644077, -5237620119669]\) \(29489595518609351/109830613939935000\) \(-11852718106456257268539735000\) \([2]\) \(311427072\) \(4.0658\)  

Rank

sage: E.rank()
 

The elliptic curves in class 333270.ea have rank \(1\).

Complex multiplication

The elliptic curves in class 333270.ea do not have complex multiplication.

Modular form 333270.2.a.ea

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - q^{10} + 4q^{11} + 6q^{13} + q^{14} + q^{16} + 2q^{17} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.