# Properties

 Label 13872b Number of curves $6$ Conductor $13872$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 13872b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13872.p5 13872b1 $$[0, -1, 0, 193, -1338]$$ $$2048/3$$ $$-1158603312$$ $$[2]$$ $$5120$$ $$0.42468$$ $$\Gamma_0(N)$$-optimal
13872.p4 13872b2 $$[0, -1, 0, -1252, -12320]$$ $$35152/9$$ $$55612958976$$ $$[2, 2]$$ $$10240$$ $$0.77125$$
13872.p2 13872b3 $$[0, -1, 0, -18592, -969488]$$ $$28756228/3$$ $$74150611968$$ $$[2]$$ $$20480$$ $$1.1178$$
13872.p3 13872b4 $$[0, -1, 0, -7032, 218880]$$ $$1556068/81$$ $$2002066523136$$ $$[2, 2]$$ $$20480$$ $$1.1178$$
13872.p1 13872b5 $$[0, -1, 0, -111072, 14285088]$$ $$3065617154/9$$ $$444903671808$$ $$[2]$$ $$40960$$ $$1.4644$$
13872.p6 13872b6 $$[0, -1, 0, 4528, 856992]$$ $$207646/6561$$ $$-324334776748032$$ $$[2]$$ $$40960$$ $$1.4644$$

## Rank

sage: E.rank()

The elliptic curves in class 13872b have rank $$1$$.

## Complex multiplication

The elliptic curves in class 13872b do not have complex multiplication.

## Modular form 13872.2.a.b

sage: E.q_eigenform(10)

$$q - q^{3} + 2q^{5} + q^{9} + 4q^{11} - 2q^{13} - 2q^{15} + 4q^{19} + O(q^{20})$$

## 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.