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SageMath
E = EllipticCurve("hg1")
E.isogeny_class()
Elliptic curves in class 227850hg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
227850.i2 | 227850hg1 | \([1, 1, 0, -582075, 185932125]\) | \(-32556488727491/3510632448\) | \(-2351849472000000000\) | \([2]\) | \(4730880\) | \(2.2634\) | \(\Gamma_0(N)\)-optimal |
227850.i1 | 227850hg2 | \([1, 1, 0, -9542075, 11341132125]\) | \(143426070964902851/1434765312\) | \(961180668000000000\) | \([2]\) | \(9461760\) | \(2.6099\) |
Rank
sage: E.rank()
The elliptic curves in class 227850hg have rank \(1\).
Complex multiplication
The elliptic curves in class 227850hg do not have complex multiplication.Modular form 227850.2.a.hg
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.