Properties

 Label 1638.b Number of curves $2$ Conductor $1638$ CM no Rank $1$ Graph Related objects

Show commands: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

Elliptic curves in class 1638.b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1638.b1 1638d1 $$[1, -1, 0, -21, 63]$$ $$-38958219/30758$$ $$-830466$$ $$$$ $$288$$ $$-0.16566$$ $$\Gamma_0(N)$$-optimal
1638.b2 1638d2 $$[1, -1, 0, 174, -964]$$ $$29503629/35672$$ $$-702131976$$ $$[]$$ $$864$$ $$0.38364$$

Rank

sage: E.rank()

The elliptic curves in class 1638.b have rank $$1$$.

Complex multiplication

The elliptic curves in class 1638.b do not have complex multiplication.

Modular form1638.2.a.b

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - 3 q^{5} + q^{7} - q^{8} + 3 q^{10} - 3 q^{11} + q^{13} - q^{14} + q^{16} + 3 q^{17} + 5 q^{19} + O(q^{20})$$ 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 LMFDB numbering.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with LMFDB labels. 