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SageMath
E = EllipticCurve("he1")
E.isogeny_class()
Elliptic curves in class 177870he
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177870.bg3 | 177870he1 | \([1, 1, 0, -1903332, -773360496]\) | \(3658671062929/880165440\) | \(183446175876881924160\) | \([2]\) | \(9953280\) | \(2.5993\) | \(\Gamma_0(N)\)-optimal |
| 177870.bg4 | 177870he2 | \([1, 1, 0, 4499988, -4854836664]\) | \(48351870250991/76871856600\) | \(-16021815314432303237400\) | \([2]\) | \(19906560\) | \(2.9459\) | |
| 177870.bg1 | 177870he3 | \([1, 1, 0, -143843592, -664084101204]\) | \(1579250141304807889/41926500\) | \(8738420918801458500\) | \([2]\) | \(29859840\) | \(3.1486\) | |
| 177870.bg2 | 177870he4 | \([1, 1, 0, -143665722, -665808123966]\) | \(-1573398910560073969/8138108343750\) | \(-1696163910426524767593750\) | \([2]\) | \(59719680\) | \(3.4952\) |
Rank
sage: E.rank()
The elliptic curves in class 177870he have rank \(0\).
Complex multiplication
The elliptic curves in class 177870he do not have complex multiplication.Modular form 177870.2.a.he
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
