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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 3850.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3850.k1 | 3850d4 | \([1, 1, 0, -652900, -197575500]\) | \(1969902499564819009/63690429687500\) | \(995162963867187500\) | \([2]\) | \(82944\) | \(2.2274\) | |
| 3850.k2 | 3850d2 | \([1, 1, 0, -89400, 10160000]\) | \(5057359576472449/51765560000\) | \(808836875000000\) | \([2]\) | \(27648\) | \(1.6781\) | |
| 3850.k3 | 3850d1 | \([1, 1, 0, -1400, 392000]\) | \(-19443408769/4249907200\) | \(-66404800000000\) | \([2]\) | \(13824\) | \(1.3315\) | \(\Gamma_0(N)\)-optimal |
| 3850.k4 | 3850d3 | \([1, 1, 0, 12600, -10570000]\) | \(14156681599871/3100231750000\) | \(-48441121093750000\) | \([2]\) | \(41472\) | \(1.8808\) |
Rank
sage: E.rank()
The elliptic curves in class 3850.k have rank \(1\).
Complex multiplication
The elliptic curves in class 3850.k do not have complex multiplication.Modular form 3850.2.a.k
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
