Properties

Label 1800s
Number of curves $6$
Conductor $1800$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("s1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 1800s

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1800.m5 1800s1 \([0, 0, 0, 150, -875]\) \(2048/3\) \(-546750000\) \([2]\) \(512\) \(0.36210\) \(\Gamma_0(N)\)-optimal
1800.m4 1800s2 \([0, 0, 0, -975, -8750]\) \(35152/9\) \(26244000000\) \([2, 2]\) \(1024\) \(0.70867\)  
1800.m2 1800s3 \([0, 0, 0, -14475, -670250]\) \(28756228/3\) \(34992000000\) \([2]\) \(2048\) \(1.0552\)  
1800.m3 1800s4 \([0, 0, 0, -5475, 148750]\) \(1556068/81\) \(944784000000\) \([2, 2]\) \(2048\) \(1.0552\)  
1800.m1 1800s5 \([0, 0, 0, -86475, 9787750]\) \(3065617154/9\) \(209952000000\) \([2]\) \(4096\) \(1.4018\)  
1800.m6 1800s6 \([0, 0, 0, 3525, 589750]\) \(207646/6561\) \(-153055008000000\) \([2]\) \(4096\) \(1.4018\)  

Rank

sage: E.rank()
 

The elliptic curves in class 1800s have rank \(1\).

Complex multiplication

The elliptic curves in class 1800s do not have complex multiplication.

Modular form 1800.2.a.s

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

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

Isogeny graph

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

The vertices are labelled with Cremona labels.