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SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 19074.bj
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 19074.bj1 | 19074bg5 | \([1, 0, 0, -114138822, -469360992672]\) | \(6812873765474836663297/74052\) | \(1787435259588\) | \([2]\) | \(1179648\) | \(2.8545\) | |
| 19074.bj2 | 19074bg3 | \([1, 0, 0, -7133682, -7334199180]\) | \(1663303207415737537/5483698704\) | \(132363155843010576\) | \([2, 2]\) | \(589824\) | \(2.5080\) | |
| 19074.bj3 | 19074bg6 | \([1, 0, 0, -7035422, -7546028088]\) | \(-1595514095015181697/95635786040388\) | \(-2308415384419102136772\) | \([4]\) | \(1179648\) | \(2.8545\) | |
| 19074.bj4 | 19074bg2 | \([1, 0, 0, -452002, -111303100]\) | \(423108074414017/23284318464\) | \(562026843542774016\) | \([2, 2]\) | \(294912\) | \(2.1614\) | |
| 19074.bj5 | 19074bg1 | \([1, 0, 0, -82082, 6849348]\) | \(2533811507137/625016832\) | \(15086386908561408\) | \([2]\) | \(147456\) | \(1.8148\) | \(\Gamma_0(N)\)-optimal |
| 19074.bj6 | 19074bg4 | \([1, 0, 0, 310958, -448684012]\) | \(137763859017023/3683199928848\) | \(-88903492423363690512\) | \([2]\) | \(589824\) | \(2.5080\) |
Rank
sage: E.rank()
The elliptic curves in class 19074.bj have rank \(0\).
Complex multiplication
The elliptic curves in class 19074.bj do not have complex multiplication.Modular form 19074.2.a.bj
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
