Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 6270d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6270.d1 | 6270d1 | \([1, 1, 0, -2757, 49401]\) | \(2318889846161881/238064062500\) | \(238064062500\) | \([2]\) | \(7680\) | \(0.91852\) | \(\Gamma_0(N)\)-optimal |
6270.d2 | 6270d2 | \([1, 1, 0, 3493, 248151]\) | \(4711131042738119/29017342901250\) | \(-29017342901250\) | \([2]\) | \(15360\) | \(1.2651\) |
Rank
sage: E.rank()
The elliptic curves in class 6270d have rank \(1\).
Complex multiplication
The elliptic curves in class 6270d do not have complex multiplication.Modular form 6270.2.a.d
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.