Show commands:
SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 870.h
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
870.h1 | 870i4 | \([1, 0, 0, -2136610, -1202204578]\) | \(1078697059648930939019041/63106084995030150\) | \(63106084995030150\) | \([2]\) | \(16000\) | \(2.2862\) | |
870.h2 | 870i3 | \([1, 0, 0, -2136580, -1202240020]\) | \(1078651622544688278688321/3692006820\) | \(3692006820\) | \([2]\) | \(8000\) | \(1.9396\) | |
870.h3 | 870i2 | \([1, 0, 0, -43360, 3450272]\) | \(9015548596898711041/63863437500000\) | \(63863437500000\) | \([10]\) | \(3200\) | \(1.4815\) | |
870.h4 | 870i1 | \([1, 0, 0, -4480, -25600]\) | \(9944061759313921/5479747200000\) | \(5479747200000\) | \([10]\) | \(1600\) | \(1.1349\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 870.h have rank \(0\).
Complex multiplication
The elliptic curves in class 870.h do not have complex multiplication.Modular form 870.2.a.h
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.