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Properties

Label 457776eu
Number of curves $4$
Conductor $457776$
CM no
Rank $1$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("eu1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 457776eu have rank \(1\).

Complex multiplication

The elliptic curves in class 457776eu do not have complex multiplication.

Modular form 457776.2.a.eu

Copy content sage:E.q_eigenform(10)
 
\(q + 2 q^{5} - q^{11} - 2 q^{13} - 8 q^{19} + O(q^{20})\) 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 457776eu

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
457776.eu3 457776eu1 \([0, 0, 0, -500259, 135844450]\) \(192100033/561\) \(40433735501254656\) \([2]\) \(4718592\) \(2.0570\) \(\Gamma_0(N)\)-optimal
457776.eu2 457776eu2 \([0, 0, 0, -708339, 12036850]\) \(545338513/314721\) \(22683325616203862016\) \([2, 2]\) \(9437184\) \(2.4035\)  
457776.eu4 457776eu3 \([0, 0, 0, 2829021, 96226018]\) \(34741712447/20160657\) \(-1453067152708588572672\) \([2]\) \(18874368\) \(2.7501\)  
457776.eu1 457776eu4 \([0, 0, 0, -7574979, -7995838718]\) \(666940371553/2756193\) \(198650942517664124928\) \([2]\) \(18874368\) \(2.7501\)