# Properties

 Label 10051b Number of curves $3$ Conductor $10051$ CM no Rank $2$ Graph # Learn more about

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

sage: E.isogeny_class()

## Elliptic curves in class 10051b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
10051.c3 10051b1 $$[0, 1, 1, 353, 230]$$ $$32768/19$$ $$-2812681891$$ $$[]$$ $$4224$$ $$0.50257$$ $$\Gamma_0(N)$$-optimal
10051.c2 10051b2 $$[0, 1, 1, -4937, 140415]$$ $$-89915392/6859$$ $$-1015378162651$$ $$[]$$ $$12672$$ $$1.0519$$
10051.c1 10051b3 $$[0, 1, 1, -406977, 99796080]$$ $$-50357871050752/19$$ $$-2812681891$$ $$[]$$ $$38016$$ $$1.6012$$

## Rank

sage: E.rank()

The elliptic curves in class 10051b have rank $$2$$.

## Complex multiplication

The elliptic curves in class 10051b do not have complex multiplication.

## Modular form 10051.2.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. 