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SageMath
E = EllipticCurve("bt1")
E.isogeny_class()
Elliptic curves in class 30030.bt
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 30030.bt1 | 30030bt8 | \([1, 0, 0, -195003991, 1045763011475]\) | \(820076206880893214178646273009/2122496008872985839843750\) | \(2122496008872985839843750\) | \([2]\) | \(9289728\) | \(3.5437\) | |
| 30030.bt2 | 30030bt5 | \([1, 0, 0, -194882701, 1047131714681]\) | \(818546927584539194367471866449/14273634375000\) | \(14273634375000\) | \([6]\) | \(3096576\) | \(2.9944\) | |
| 30030.bt3 | 30030bt7 | \([1, 0, 0, -177737371, -908394585049]\) | \(620954771108295351491118574129/2882378618771462717156250\) | \(2882378618771462717156250\) | \([2]\) | \(9289728\) | \(3.5437\) | |
| 30030.bt4 | 30030bt6 | \([1, 0, 0, -16956121, 2366883701]\) | \(539142086340577084766074129/309580507925165039062500\) | \(309580507925165039062500\) | \([2, 2]\) | \(4644864\) | \(3.1971\) | |
| 30030.bt5 | 30030bt4 | \([1, 0, 0, -12369181, 15826636745]\) | \(209289070072300727183442769/12893854589717635333800\) | \(12893854589717635333800\) | \([6]\) | \(3096576\) | \(2.9944\) | |
| 30030.bt6 | 30030bt2 | \([1, 0, 0, -12180181, 16360637345]\) | \(199841159336796255944706769/834505270358760000\) | \(834505270358760000\) | \([2, 6]\) | \(1548288\) | \(2.6478\) | |
| 30030.bt7 | 30030bt1 | \([1, 0, 0, -749461, 263897441]\) | \(-46555485820017544148689/3157693080314572800\) | \(-3157693080314572800\) | \([12]\) | \(774144\) | \(2.3012\) | \(\Gamma_0(N)\)-optimal |
| 30030.bt8 | 30030bt3 | \([1, 0, 0, 4220699, 295790705]\) | \(8315279469612171276463151/4849789796887785750000\) | \(-4849789796887785750000\) | \([4]\) | \(2322432\) | \(2.8505\) |
Rank
sage: E.rank()
The elliptic curves in class 30030.bt have rank \(1\).
Complex multiplication
The elliptic curves in class 30030.bt do not have complex multiplication.Modular form 30030.2.a.bt
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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 4 & 2 & 12 & 6 & 12 & 4 \\ 3 & 1 & 12 & 6 & 4 & 2 & 4 & 12 \\ 4 & 12 & 1 & 2 & 3 & 6 & 12 & 4 \\ 2 & 6 & 2 & 1 & 6 & 3 & 6 & 2 \\ 12 & 4 & 3 & 6 & 1 & 2 & 4 & 12 \\ 6 & 2 & 6 & 3 & 2 & 1 & 2 & 6 \\ 12 & 4 & 12 & 6 & 4 & 2 & 1 & 3 \\ 4 & 12 & 4 & 2 & 12 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
