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Properties

Label 470400ew
Number of curves $2$
Conductor $470400$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ew1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 470400ew have rank \(0\).

Complex multiplication

The elliptic curves in class 470400ew do not have complex multiplication.

Modular form 470400.2.a.ew

Copy content sage:E.q_eigenform(10)
 
\(q - q^{3} + q^{9} + 2 q^{11} + 6 q^{13} - 8 q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 470400ew

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
470400.ew2 470400ew1 \([0, -1, 0, -5765258, -5397638238]\) \(-262583645216/4100625\) \(-330950019408750000000\) \([2]\) \(24772608\) \(2.7391\) \(\Gamma_0(N)\)-optimal*
470400.ew1 470400ew2 \([0, -1, 0, -92587133, -342874266363]\) \(8496758995072/2025\) \(20919309868800000000\) \([2]\) \(49545216\) \(3.0857\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 470400ew1.