Show commands:
SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 20400dk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
20400.dp2 | 20400dk1 | \([0, 1, 0, -102208, -12606412]\) | \(1845026709625/793152\) | \(50761728000000\) | \([2]\) | \(82944\) | \(1.5900\) | \(\Gamma_0(N)\)-optimal |
20400.dp3 | 20400dk2 | \([0, 1, 0, -86208, -16670412]\) | \(-1107111813625/1228691592\) | \(-78636261888000000\) | \([2]\) | \(165888\) | \(1.9366\) | |
20400.dp1 | 20400dk3 | \([0, 1, 0, -300208, 47765588]\) | \(46753267515625/11591221248\) | \(741838159872000000\) | \([2]\) | \(248832\) | \(2.1393\) | |
20400.dp4 | 20400dk4 | \([0, 1, 0, 723792, 303765588]\) | \(655215969476375/1001033261568\) | \(-64066128740352000000\) | \([2]\) | \(497664\) | \(2.4859\) |
Rank
sage: E.rank()
The elliptic curves in class 20400dk have rank \(0\).
Complex multiplication
The elliptic curves in class 20400dk do not have complex multiplication.Modular form 20400.2.a.dk
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.