Show commands:
SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 390390bz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 390390.bz7 | 390390bz1 | \([1, 0, 1, -126658913, 579909336788]\) | \(-46555485820017544148689/3157693080314572800\) | \(-15241581379300102822195200\) | \([2]\) | \(130056192\) | \(3.5837\) | \(\Gamma_0(N)\)-optimal |
| 390390.bz6 | 390390bz2 | \([1, 0, 1, -2058450593, 35946378697556]\) | \(199841159336796255944706769/834505270358760000\) | \(4027997549515095996840000\) | \([2, 2]\) | \(260112384\) | \(3.9303\) | |
| 390390.bz8 | 390390bz3 | \([1, 0, 1, 713298127, 649138880756]\) | \(8315279469612171276463151/4849789796887785750000\) | \(-23409009039726136248171750000\) | \([2]\) | \(390168576\) | \(4.1330\) | |
| 390390.bz2 | 390390bz4 | \([1, 0, 1, -32935176473, 2300581312330628]\) | \(818546927584539194367471866449/14273634375000\) | \(68896106863959375000\) | \([2]\) | \(520224768\) | \(4.2768\) | |
| 390390.bz5 | 390390bz5 | \([1, 0, 1, -2090391593, 34773211320356]\) | \(209289070072300727183442769/12893854589717635333800\) | \(62236173378340389687903844200\) | \([2]\) | \(520224768\) | \(4.2768\) | |
| 390390.bz4 | 390390bz6 | \([1, 0, 1, -2865584453, 5202909075548]\) | \(539142086340577084766074129/309580507925165039062500\) | \(1494285981877757937032226562500\) | \([2, 2]\) | \(780337152\) | \(4.4796\) | |
| 390390.bz1 | 390390bz7 | \([1, 0, 1, -32955674483, 2297574291885056]\) | \(820076206880893214178646273009/2122496008872985839843750\) | \(10244882838092207908630371093750\) | \([2]\) | \(1560674304\) | \(4.8261\) | |
| 390390.bz3 | 390390bz8 | \([1, 0, 1, -30037615703, -1995712865736952]\) | \(620954771108295351491118574129/2882378618771462717156250\) | \(13912691058493665186334241906250\) | \([2]\) | \(1560674304\) | \(4.8261\) |
Rank
sage: E.rank()
The elliptic curves in class 390390bz have rank \(0\).
Complex multiplication
The elliptic curves in class 390390bz do not have complex multiplication.Modular form 390390.2.a.bz
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.
