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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 2850.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2850.j1 | 2850j3 | \([1, 0, 1, -2188801, -1246582252]\) | \(74220219816682217473/16416\) | \(256500000\) | \([2]\) | \(30720\) | \(1.9062\) | |
2850.j2 | 2850j2 | \([1, 0, 1, -136801, -19486252]\) | \(18120364883707393/269485056\) | \(4210704000000\) | \([2, 2]\) | \(15360\) | \(1.5596\) | |
2850.j3 | 2850j4 | \([1, 0, 1, -132801, -20678252]\) | \(-16576888679672833/2216253521952\) | \(-34628961280500000\) | \([2]\) | \(30720\) | \(1.9062\) | |
2850.j4 | 2850j1 | \([1, 0, 1, -8801, -286252]\) | \(4824238966273/537919488\) | \(8404992000000\) | \([2]\) | \(7680\) | \(1.2130\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2850.j have rank \(0\).
Complex multiplication
The elliptic curves in class 2850.j do not have complex multiplication.Modular form 2850.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.