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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 13860.k
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 13860.k1 | 13860c3 | \([0, 0, 0, -9288, 27837]\) | \(281370820608/161767375\) | \(50945075874000\) | \([2]\) | \(31104\) | \(1.3198\) | |
| 13860.k2 | 13860c1 | \([0, 0, 0, -6648, 208633]\) | \(75216478666752/326095\) | \(140873040\) | \([6]\) | \(10368\) | \(0.77046\) | \(\Gamma_0(N)\)-optimal |
| 13860.k3 | 13860c2 | \([0, 0, 0, -6543, 215542]\) | \(-4481782160112/310023175\) | \(-2142880185600\) | \([6]\) | \(20736\) | \(1.1170\) | |
| 13860.k4 | 13860c4 | \([0, 0, 0, 37017, 222318]\) | \(1113258734352/648484375\) | \(-3267614196000000\) | \([2]\) | \(62208\) | \(1.6663\) |
Rank
sage: E.rank()
The elliptic curves in class 13860.k have rank \(0\).
Complex multiplication
The elliptic curves in class 13860.k do not have complex multiplication.Modular form 13860.2.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.
