Properties

Label 428400lv
Number of curves $4$
Conductor $428400$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("lv1")
 
E.isogeny_class()
 

Elliptic curves in class 428400lv

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
428400.lv4 428400lv1 \([0, 0, 0, 50325, 7308250]\) \(302111711/669375\) \(-31230360000000000\) \([2]\) \(3145728\) \(1.8492\) \(\Gamma_0(N)\)-optimal*
428400.lv3 428400lv2 \([0, 0, 0, -399675, 79758250]\) \(151334226289/28676025\) \(1337908622400000000\) \([2, 2]\) \(6291456\) \(2.1957\) \(\Gamma_0(N)\)-optimal*
428400.lv1 428400lv3 \([0, 0, 0, -6069675, 5755428250]\) \(530044731605089/26309115\) \(1227478069440000000\) \([2]\) \(12582912\) \(2.5423\) \(\Gamma_0(N)\)-optimal*
428400.lv2 428400lv4 \([0, 0, 0, -1929675, -959111750]\) \(17032120495489/1339001685\) \(62472462615360000000\) \([2]\) \(12582912\) \(2.5423\)  
*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 428400lv1.

Rank

sage: E.rank()
 

The elliptic curves in class 428400lv have rank \(0\).

Complex multiplication

The elliptic curves in class 428400lv do not have complex multiplication.

Modular form 428400.2.a.lv

sage: E.q_eigenform(10)
 
\(q + q^{7} + 6 q^{13} - q^{17} - 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

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

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

The vertices are labelled with Cremona labels.