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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)
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.