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SageMath
E = EllipticCurve("kb1")
E.isogeny_class()
Elliptic curves in class 381150kb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
381150.kb4 | 381150kb1 | \([1, -1, 1, 6982645, 9453621147]\) | \(1865864036231/2993760000\) | \(-60411642919897500000000\) | \([2]\) | \(29491200\) | \(3.0565\) | \(\Gamma_0(N)\)-optimal |
381150.kb3 | 381150kb2 | \([1, -1, 1, -47467355, 96464721147]\) | \(586145095611769/140040608400\) | \(2825905626685505306250000\) | \([2, 2]\) | \(58982400\) | \(3.4030\) | |
381150.kb1 | 381150kb3 | \([1, -1, 1, -709034855, 7266533286147]\) | \(1953542217204454969/170843779260\) | \(3447488572143695670937500\) | \([2]\) | \(117964800\) | \(3.7496\) | |
381150.kb2 | 381150kb4 | \([1, -1, 1, -257099855, -1505546843853]\) | \(93137706732176569/5369647977540\) | \(108355130746852209629062500\) | \([2]\) | \(117964800\) | \(3.7496\) |
Rank
sage: E.rank()
The elliptic curves in class 381150kb have rank \(0\).
Complex multiplication
The elliptic curves in class 381150kb do not have complex multiplication.Modular form 381150.2.a.kb
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.