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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 20160.bz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 20160.bz1 | 20160bn7 | \([0, 0, 0, -3715788, -1734477712]\) | \(29689921233686449/10380965400750\) | \(1983833381836357632000\) | \([2]\) | \(884736\) | \(2.7881\) | |
| 20160.bz2 | 20160bn4 | \([0, 0, 0, -3318348, -2326649488]\) | \(21145699168383889/2593080\) | \(495545305006080\) | \([2]\) | \(294912\) | \(2.2387\) | |
| 20160.bz3 | 20160bn6 | \([0, 0, 0, -1555788, 727058288]\) | \(2179252305146449/66177562500\) | \(12646729138176000000\) | \([2, 2]\) | \(442368\) | \(2.4415\) | |
| 20160.bz4 | 20160bn3 | \([0, 0, 0, -1544268, 738638192]\) | \(2131200347946769/2058000\) | \(393289924608000\) | \([2]\) | \(221184\) | \(2.0949\) | |
| 20160.bz5 | 20160bn2 | \([0, 0, 0, -207948, -36150928]\) | \(5203798902289/57153600\) | \(10922223049113600\) | \([2, 2]\) | \(147456\) | \(1.8922\) | |
| 20160.bz6 | 20160bn5 | \([0, 0, 0, -46668, -90792592]\) | \(-58818484369/18600435000\) | \(-3554598483394560000\) | \([2]\) | \(294912\) | \(2.2387\) | |
| 20160.bz7 | 20160bn1 | \([0, 0, 0, -23628, 491888]\) | \(7633736209/3870720\) | \(739706111262720\) | \([2]\) | \(73728\) | \(1.5456\) | \(\Gamma_0(N)\)-optimal |
| 20160.bz8 | 20160bn8 | \([0, 0, 0, 419892, 2447480432]\) | \(42841933504271/13565917968750\) | \(-2592487296000000000000\) | \([2]\) | \(884736\) | \(2.7881\) |
Rank
sage: E.rank()
The elliptic curves in class 20160.bz have rank \(1\).
Complex multiplication
The elliptic curves in class 20160.bz do not have complex multiplication.Modular form 20160.2.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 LMFDB numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
