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Properties

Label 414960.fp
Number of curves $4$
Conductor $414960$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("fp1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 414960.fp have rank \(0\).

Complex multiplication

The elliptic curves in class 414960.fp do not have complex multiplication.

Modular form 414960.2.a.fp

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} - q^{5} + q^{7} + q^{9} + q^{13} - q^{15} + 6 q^{17} + q^{19} + O(q^{20})\) 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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 414960.fp

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
414960.fp1 414960fp3 \([0, 1, 0, -59016536, 174485655060]\) \(5549896908024170183373529/56019600\) \(229456281600\) \([2]\) \(15728640\) \(2.6887\) \(\Gamma_0(N)\)-optimal*
414960.fp2 414960fp4 \([0, 1, 0, -3738456, 2647796244]\) \(1410719602237262088409/76269550743750000\) \(312400079846400000000\) \([2]\) \(15728640\) \(2.6887\)  
414960.fp3 414960fp2 \([0, 1, 0, -3688536, 2725411860]\) \(1354958399265695661529/4304795040000\) \(17632440483840000\) \([2, 2]\) \(7864320\) \(2.3421\) \(\Gamma_0(N)\)-optimal*
414960.fp4 414960fp1 \([0, 1, 0, -227416, 43736084]\) \(-317562142497484249/18670942617600\) \(-76476180961689600\) \([2]\) \(3932160\) \(1.9955\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 414960.fp1.