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Properties

Label 473200ha
Number of curves $2$
Conductor $473200$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("ha1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 473200ha have rank \(1\).

Complex multiplication

The elliptic curves in class 473200ha do not have complex multiplication.

Modular form 473200.2.a.ha

Copy content sage:E.q_eigenform(10)
 
\(q + 2 q^{3} - q^{7} + q^{9} + 6 q^{11} - 2 q^{17} + O(q^{20})\) 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 473200ha

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
473200.ha2 473200ha1 \([0, -1, 0, -16259208, 25048676912]\) \(12310389629/107653\) \(4156963754216000000000\) \([2]\) \(36126720\) \(2.9721\) \(\Gamma_0(N)\)-optimal
473200.ha1 473200ha2 \([0, -1, 0, -28089208, -16214363088]\) \(63473450669/33787663\) \(1304692766858936000000000\) \([2]\) \(72253440\) \(3.3187\)