Show commands:
SageMath
E = EllipticCurve("eh1")
E.isogeny_class()
Elliptic curves in class 333270.eh
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.eh1 | 333270eh3 | \([1, -1, 1, -13747487, -6448849081]\) | \(2662558086295801/1374177967680\) | \(148298762018407387222080\) | \([2]\) | \(36495360\) | \(3.1382\) | |
333270.eh2 | 333270eh1 | \([1, -1, 1, -7677212, 8189226299]\) | \(463702796512201/15214500\) | \(1641920892195874500\) | \([2]\) | \(12165120\) | \(2.5889\) | \(\Gamma_0(N)\)-optimal |
333270.eh3 | 333270eh2 | \([1, -1, 1, -7343942, 8932285091]\) | \(-405897921250921/84358968750\) | \(-9103864946907482718750\) | \([2]\) | \(24330240\) | \(2.9355\) | |
333270.eh4 | 333270eh4 | \([1, -1, 1, 51573433, -50109352009]\) | \(140574743422291079/91397357868600\) | \(-9863434971636093369156600\) | \([2]\) | \(72990720\) | \(3.4848\) |
Rank
sage: E.rank()
The elliptic curves in class 333270.eh have rank \(1\).
Complex multiplication
The elliptic curves in class 333270.eh do not have complex multiplication.Modular form 333270.2.a.eh
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.