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Properties

Label 412200.x
Number of curves $2$
Conductor $412200$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("x1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 412200.x have rank \(1\).

Complex multiplication

The elliptic curves in class 412200.x do not have complex multiplication.

Modular form 412200.2.a.x

Copy content sage:E.q_eigenform(10)
 
\(q + 2 q^{7} - 4 q^{11} - 4 q^{13} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB 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 LMFDB labels.

Elliptic curves in class 412200.x

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
412200.x1 412200x2 \([0, 0, 0, -167175, 25454250]\) \(177194479824/6555125\) \(19114744500000000\) \([2]\) \(2359296\) \(1.8932\) \(\Gamma_0(N)\)-optimal*
412200.x2 412200x1 \([0, 0, 0, -26550, -1123875]\) \(11356637184/3578125\) \(652113281250000\) \([2]\) \(1179648\) \(1.5467\) \(\Gamma_0(N)\)-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 412200.x1.