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Properties

Label 418950qc
Number of curves $2$
Conductor $418950$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("qc1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 418950qc have rank \(0\).

Complex multiplication

The elliptic curves in class 418950qc do not have complex multiplication.

Modular form 418950.2.a.qc

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{8} + 4 q^{11} + 2 q^{13} + q^{16} + 6 q^{17} + q^{19} + O(q^{20})\) 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 418950qc

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
418950.qc2 418950qc1 \([1, -1, 1, -1742180, 923085447]\) \(-3491055413/175104\) \(-29332013382000000000\) \([2]\) \(14745600\) \(2.4963\) \(\Gamma_0(N)\)-optimal*
418950.qc1 418950qc2 \([1, -1, 1, -28202180, 57653325447]\) \(14809006736693/34656\) \(5805294315187500000\) \([2]\) \(29491200\) \(2.8429\) \(\Gamma_0(N)\)-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 418950qc1.