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Properties

Label 485100.eu
Number of curves $4$
Conductor $485100$
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 485100.eu have rank \(1\).

Complex multiplication

The elliptic curves in class 485100.eu do not have complex multiplication.

Modular form 485100.2.a.eu

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

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 485100.eu

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
485100.eu1 485100eu4 \([0, 0, 0, -17158575, -14860732250]\) \(1628514404944/664335375\) \(227909872627321500000000\) \([2]\) \(47775744\) \(3.1796\)  
485100.eu2 485100eu2 \([0, 0, 0, -7897575, 8541814750]\) \(158792223184/16335\) \(5603958346140000000\) \([2]\) \(15925248\) \(2.6303\) \(\Gamma_0(N)\)-optimal*
485100.eu3 485100eu1 \([0, 0, 0, -455700, 154821625]\) \(-488095744/200475\) \(-4298490776868750000\) \([2]\) \(7962624\) \(2.2837\) \(\Gamma_0(N)\)-optimal*
485100.eu4 485100eu3 \([0, 0, 0, 3513300, -1692747875]\) \(223673040896/187171875\) \(-4013251419761718750000\) \([2]\) \(23887872\) \(2.8330\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 485100.eu1.