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SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 16830bj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
16830.bi3 | 16830bj1 | \([1, -1, 1, -360443, 83386091]\) | \(-191808834096148160787/11043434659840\) | \(-298172735815680\) | \([6]\) | \(161280\) | \(1.8408\) | \(\Gamma_0(N)\)-optimal |
16830.bi2 | 16830bj2 | \([1, -1, 1, -5767163, 5332229867]\) | \(785681552361835673854227/2604236800\) | \(70314393600\) | \([6]\) | \(322560\) | \(2.1874\) | |
16830.bi4 | 16830bj3 | \([1, -1, 1, -37883, 225326827]\) | \(-305460292990923/1114070936704000\) | \(-21928258247144832000\) | \([2]\) | \(483840\) | \(2.3901\) | |
16830.bi1 | 16830bj4 | \([1, -1, 1, -5787803, 5292156331]\) | \(1089365384367428097483/16063552169500000\) | \(316178897352268500000\) | \([2]\) | \(967680\) | \(2.7367\) |
Rank
sage: E.rank()
The elliptic curves in class 16830bj have rank \(0\).
Complex multiplication
The elliptic curves in class 16830bj do not have complex multiplication.Modular form 16830.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.