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Properties

Label 486720mt
Number of curves $4$
Conductor $486720$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("mt1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 486720mt have rank \(1\).

Complex multiplication

The elliptic curves in class 486720mt do not have complex multiplication.

Modular form 486720.2.a.mt

Copy content sage:E.q_eigenform(10)
 
\(q + q^{5} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 486720mt

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
486720.mt3 486720mt1 \([0, 0, 0, -395967, 95903444]\) \(30488290624/195\) \(43913922137280\) \([2]\) \(2064384\) \(1.8024\) \(\Gamma_0(N)\)-optimal
486720.mt2 486720mt2 \([0, 0, 0, -403572, 92027936]\) \(504358336/38025\) \(548045748273254400\) \([2, 2]\) \(4128768\) \(2.1490\)  
486720.mt4 486720mt3 \([0, 0, 0, 387348, 408712304]\) \(55742968/658125\) \(-75883257453219840000\) \([2]\) \(8257536\) \(2.4956\)  
486720.mt1 486720mt4 \([0, 0, 0, -1316172, -472688944]\) \(2186875592/428415\) \(49397190111029329920\) \([2]\) \(8257536\) \(2.4956\)