Show commands:
SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 161874.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
161874.g1 | 161874bp6 | \([1, -1, 0, -132089283, -584285278581]\) | \(2361739090258884097/5202\) | \(561390284347362\) | \([2]\) | \(12976128\) | \(2.9641\) | |
161874.g2 | 161874bp4 | \([1, -1, 0, -8255673, -9127693575]\) | \(576615941610337/27060804\) | \(2920352259174977124\) | \([2, 2]\) | \(6488064\) | \(2.6176\) | |
161874.g3 | 161874bp5 | \([1, -1, 0, -7827183, -10117762569]\) | \(-491411892194497/125563633938\) | \(-13550596724364070242978\) | \([2]\) | \(12976128\) | \(2.9641\) | |
161874.g4 | 161874bp2 | \([1, -1, 0, -542853, -126832635]\) | \(163936758817/30338064\) | \(3274028138313815184\) | \([2, 2]\) | \(3244032\) | \(2.2710\) | |
161874.g5 | 161874bp1 | \([1, -1, 0, -161973, 23310261]\) | \(4354703137/352512\) | \(38042447504009472\) | \([2]\) | \(1622016\) | \(1.9244\) | \(\Gamma_0(N)\)-optimal |
161874.g6 | 161874bp3 | \([1, -1, 0, 1075887, -739687599]\) | \(1276229915423/2927177028\) | \(-315895567874660903268\) | \([2]\) | \(6488064\) | \(2.6176\) |
Rank
sage: E.rank()
The elliptic curves in class 161874.g have rank \(0\).
Complex multiplication
The elliptic curves in class 161874.g do not have complex multiplication.Modular form 161874.2.a.g
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.