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Properties

Label 333270.eu
Number of curves $2$
Conductor $333270$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("eu1")
 
E.isogeny_class()
 

Elliptic curves in class 333270.eu

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
333270.eu1 333270eu1 \([1, -1, 1, -24396257, 46286108209]\) \(551105805571803/1376829440\) \(4011792409097331425280\) \([2]\) \(42577920\) \(3.0227\) \(\Gamma_0(N)\)-optimal
333270.eu2 333270eu2 \([1, -1, 1, -15255137, 81384352561]\) \(-134745327251163/903920796800\) \(-2633835742956986299401600\) \([2]\) \(85155840\) \(3.3692\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 333270.2.a.eu

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + q^{10} - 4 q^{11} + 6 q^{13} + q^{14} + q^{16} + 4 q^{17} - 6 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.