Show commands:
SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 6270m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6270.m4 | 6270m1 | \([1, 1, 1, 44, 149]\) | \(9407293631/12840960\) | \(-12840960\) | \([2]\) | \(1536\) | \(0.049737\) | \(\Gamma_0(N)\)-optimal |
6270.m3 | 6270m2 | \([1, 1, 1, -276, 1173]\) | \(2325676477249/629006400\) | \(629006400\) | \([2, 2]\) | \(3072\) | \(0.39631\) | |
6270.m2 | 6270m3 | \([1, 1, 1, -1596, -24171]\) | \(449613538734529/21502965000\) | \(21502965000\) | \([2]\) | \(6144\) | \(0.74288\) | |
6270.m1 | 6270m4 | \([1, 1, 1, -4076, 98453]\) | \(7489156350944449/901299960\) | \(901299960\) | \([2]\) | \(6144\) | \(0.74288\) |
Rank
sage: E.rank()
The elliptic curves in class 6270m have rank \(1\).
Complex multiplication
The elliptic curves in class 6270m do not have complex multiplication.Modular form 6270.2.a.m
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.