Properties

Label 90090cd
Number of curves $8$
Conductor $90090$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("cd1")
 
E.isogeny_class()
 

Elliptic curves in class 90090cd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
90090.ce7 90090cd1 \([1, -1, 0, -6745149, -7125230907]\) \(-46555485820017544148689/3157693080314572800\) \(-2301958255549323571200\) \([2]\) \(6193152\) \(2.8505\) \(\Gamma_0(N)\)-optimal
90090.ce6 90090cd2 \([1, -1, 0, -109621629, -441737208315]\) \(199841159336796255944706769/834505270358760000\) \(608354342091536040000\) \([2, 2]\) \(12386304\) \(3.1971\)  
90090.ce8 90090cd3 \([1, -1, 0, 37986291, -7986349035]\) \(8315279469612171276463151/4849789796887785750000\) \(-3535496761931195811750000\) \([6]\) \(18579456\) \(3.3998\)  
90090.ce5 90090cd4 \([1, -1, 0, -111322629, -427319192115]\) \(209289070072300727183442769/12893854589717635333800\) \(9399619995904156158340200\) \([2]\) \(24772608\) \(3.5437\)  
90090.ce2 90090cd5 \([1, -1, 0, -1753944309, -28272556296387]\) \(818546927584539194367471866449/14273634375000\) \(10405479459375000\) \([2]\) \(24772608\) \(3.5437\)  
90090.ce4 90090cd6 \([1, -1, 0, -152605089, -63905859927]\) \(539142086340577084766074129/309580507925165039062500\) \(225684190277445313476562500\) \([2, 6]\) \(37158912\) \(3.7464\)  
90090.ce3 90090cd7 \([1, -1, 0, -1599636339, 24526653796323]\) \(620954771108295351491118574129/2882378618771462717156250\) \(2101254013084396320806906250\) \([6]\) \(74317824\) \(4.0930\)  
90090.ce1 90090cd8 \([1, -1, 0, -1755035919, -28235601309825]\) \(820076206880893214178646273009/2122496008872985839843750\) \(1547299590468406677246093750\) \([6]\) \(74317824\) \(4.0930\)  

Rank

sage: E.rank()
 

The elliptic curves in class 90090cd have rank \(0\).

Complex multiplication

The elliptic curves in class 90090cd do not have complex multiplication.

Modular form 90090.2.a.cd

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - q^{10} + q^{11} + q^{13} - q^{14} + q^{16} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.