Properties

 Label 177870hp Number of curves $8$ Conductor $177870$ CM no Rank $0$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("177870.bs1")

sage: E.isogeny_class()

Elliptic curves in class 177870hp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
177870.bs7 177870hp1 [1, 1, 0, -243212, -16345776] [2] 3317760 $$\Gamma_0(N)$$-optimal
177870.bs5 177870hp2 [1, 1, 0, -2140492, 1192980496] [2, 2] 6635520
177870.bs4 177870hp3 [1, 1, 0, -15895772, -24399903744] [2] 9953280
177870.bs2 177870hp4 [1, 1, 0, -34157092, 76822593016] [2] 13271040
177870.bs6 177870hp5 [1, 1, 0, -480372, 2998194984] [2] 13271040
177870.bs3 177870hp6 [1, 1, 0, -16014352, -24017530676] [2, 2] 19906560
177870.bs1 177870hp7 [1, 1, 0, -38248102, 57264612574] [2] 39813120
177870.bs8 177870hp8 [1, 1, 0, 4322118, -80825425974] [2] 39813120

Rank

sage: E.rank()

The elliptic curves in class 177870hp have rank $$0$$.

Modular form 177870.2.a.bs

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} - q^{8} + q^{9} - q^{10} - q^{12} + 2q^{13} - q^{15} + q^{16} - 6q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with Cremona labels.