Properties

 Label 37200cz Number of curves $6$ Conductor $37200$ CM no Rank $0$ Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("37200.dc1")

sage: E.isogeny_class()

Elliptic curves in class 37200cz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
37200.dc6 37200cz1 [0, 1, 0, 23992, 8111988] [2] 294912 $$\Gamma_0(N)$$-optimal
37200.dc5 37200cz2 [0, 1, 0, -488008, 123823988] [2, 2] 589824
37200.dc4 37200cz3 [0, 1, 0, -1480008, -540816012] [2] 1179648
37200.dc2 37200cz4 [0, 1, 0, -7688008, 8202223988] [2, 2] 1179648
37200.dc3 37200cz5 [0, 1, 0, -7568008, 8470783988] [2] 2359296
37200.dc1 37200cz6 [0, 1, 0, -123008008, 525066463988] [2] 2359296

Rank

sage: E.rank()

The elliptic curves in class 37200cz have rank $$0$$.

Modular form 37200.2.a.dc

sage: E.q_eigenform(10)

$$q + q^{3} + q^{9} + 4q^{11} - 6q^{13} - 2q^{17} - 4q^{19} + O(q^{20})$$

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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels.