Properties

Label 428400.it
Number of curves $6$
Conductor $428400$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("it1")
 
E.isogeny_class()
 

Elliptic curves in class 428400.it

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
428400.it1 428400it6 \([0, 0, 0, -258264075, -1594414079750]\) \(40832710302042509761/91556816413125\) \(4271674826570760000000000\) \([2]\) \(100663296\) \(3.6084\)  
428400.it2 428400it4 \([0, 0, 0, -22014075, -5160329750]\) \(25288177725059761/14387797265625\) \(671277069225000000000000\) \([2, 2]\) \(50331648\) \(3.2618\)  
428400.it3 428400it2 \([0, 0, 0, -14076075, 20233332250]\) \(6610905152742241/35128130625\) \(1638938062440000000000\) \([2, 2]\) \(25165824\) \(2.9152\)  
428400.it4 428400it1 \([0, 0, 0, -14058075, 20287890250]\) \(6585576176607121/187425\) \(8744500800000000\) \([2]\) \(12582912\) \(2.5687\) \(\Gamma_0(N)\)-optimal*
428400.it5 428400it3 \([0, 0, 0, -6426075, 42135282250]\) \(-629004249876241/16074715228425\) \(-749981913697396800000000\) \([2]\) \(50331648\) \(3.2618\)  
428400.it6 428400it5 \([0, 0, 0, 87227925, -41100947750]\) \(1573196002879828319/926055908203125\) \(-43206064453125000000000000\) \([2]\) \(100663296\) \(3.6084\)  
*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 428400.it1.

Rank

sage: E.rank()
 

The elliptic curves in class 428400.it have rank \(1\).

Complex multiplication

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

Modular form 428400.2.a.it

sage: E.q_eigenform(10)
 
\(q + q^{7} - 4 q^{11} + 2 q^{13} + q^{17} + 4 q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.