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Properties

Label 462462.t
Number of curves $4$
Conductor $462462$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("t1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 462462.t have rank \(0\).

Complex multiplication

The elliptic curves in class 462462.t do not have complex multiplication.

Modular form 462462.2.a.t

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - 2 q^{5} + q^{6} - q^{8} + q^{9} + 2 q^{10} - q^{12} + q^{13} + 2 q^{15} + q^{16} + 6 q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 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 462462.t

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
462462.t1 462462t3 \([1, 1, 0, -54558781, 155073778105]\) \(86173423834915873/9892398516\) \(2061797243493611547924\) \([2]\) \(70778880\) \(3.1169\) \(\Gamma_0(N)\)-optimal*
462462.t2 462462t4 \([1, 1, 0, -21119221, -35742169679]\) \(4998193642364833/245879021388\) \(51246690851641096343532\) \([2]\) \(70778880\) \(3.1169\)  
462462.t3 462462t2 \([1, 1, 0, -3687961, 2003480725]\) \(26615737485793/7070119056\) \(1473571041164113875984\) \([2, 2]\) \(35389440\) \(2.7703\) \(\Gamma_0(N)\)-optimal*
462462.t4 462462t1 \([1, 1, 0, 580919, 202867141]\) \(104021936927/145297152\) \(-30283178239993206528\) \([2]\) \(17694720\) \(2.4237\) \(\Gamma_0(N)\)-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 462462.t1.