Show commands:
SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 6699d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6699.a4 | 6699d1 | \([1, 1, 1, -33814, 2404610]\) | \(-4275768267198290017/52843101620463\) | \(-52843101620463\) | \([4]\) | \(20736\) | \(1.4439\) | \(\Gamma_0(N)\)-optimal |
6699.a3 | 6699d2 | \([1, 1, 1, -542619, 153621456]\) | \(17668869054438249282097/2649918412449\) | \(2649918412449\) | \([2, 2]\) | \(41472\) | \(1.7905\) | |
6699.a2 | 6699d3 | \([1, 1, 1, -544214, 152670836]\) | \(17825137625614555960417/216318148151991039\) | \(216318148151991039\) | \([2]\) | \(82944\) | \(2.1370\) | |
6699.a1 | 6699d4 | \([1, 1, 1, -8681904, 9842626320]\) | \(72371679832051361738355457/1627857\) | \(1627857\) | \([2]\) | \(82944\) | \(2.1370\) |
Rank
sage: E.rank()
The elliptic curves in class 6699d have rank \(1\).
Complex multiplication
The elliptic curves in class 6699d do not have complex multiplication.Modular form 6699.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.