Properties

Label 418950mc
Number of curves $4$
Conductor $418950$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("mc1")
 
E.isogeny_class()
 

Elliptic curves in class 418950mc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
418950.mc4 418950mc1 \([1, -1, 1, -20947730, 38573658897]\) \(-758575480593601/40535043840\) \(-54320835542527260000000\) \([2]\) \(53084160\) \(3.1213\) \(\Gamma_0(N)\)-optimal*
418950.mc3 418950mc2 \([1, -1, 1, -339349730, 2406210930897]\) \(3225005357698077121/8526675600\) \(11426560800583556250000\) \([2, 2]\) \(106168320\) \(3.4679\) \(\Gamma_0(N)\)-optimal*
418950.mc1 418950mc3 \([1, -1, 1, -5429592230, 153993632580897]\) \(13209596798923694545921/92340\) \(123744431455312500\) \([2]\) \(212336640\) \(3.8145\) \(\Gamma_0(N)\)-optimal*
418950.mc2 418950mc4 \([1, -1, 1, -343539230, 2343753864897]\) \(3345930611358906241/165622259047500\) \(221949667340019214804687500\) \([2]\) \(212336640\) \(3.8145\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 418950mc1.

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 418950.2.a.mc

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.