# Properties

 Label 31939b Number of curves $3$ Conductor $31939$ CM no Rank $1$ Graph # Learn more about

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

sage: E.isogeny_class()

## Elliptic curves in class 31939b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
31939.e3 31939b1 $$[0, -1, 1, 1121, -1012]$$ $$32768/19$$ $$-90251980579$$ $$[]$$ $$23040$$ $$0.79161$$ $$\Gamma_0(N)$$-optimal
31939.e2 31939b2 $$[0, -1, 1, -15689, -799487]$$ $$-89915392/6859$$ $$-32580964989019$$ $$[]$$ $$69120$$ $$1.3409$$
31939.e1 31939b3 $$[0, -1, 1, -1293249, -565640702]$$ $$-50357871050752/19$$ $$-90251980579$$ $$[]$$ $$207360$$ $$1.8902$$

## Rank

sage: E.rank()

The elliptic curves in class 31939b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 31939b do not have complex multiplication.

## Modular form 31939.2.a.b

sage: E.q_eigenform(10)

$$q + 2q^{3} - 2q^{4} + 3q^{5} + q^{7} + q^{9} - 3q^{11} - 4q^{12} + 4q^{13} + 6q^{15} + 4q^{16} + 3q^{17} - q^{19} + O(q^{20})$$ ## 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 Cremona numbering.

$$\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 