# Properties

 Label 31200bp Number of curves $2$ Conductor $31200$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("bp1")

sage: E.isogeny_class()

## Elliptic curves in class 31200bp

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
31200.c1 31200bp1 $$[0, -1, 0, -72558, -7496388]$$ $$42246001231552/14414517$$ $$14414517000000$$ $$[2]$$ $$98304$$ $$1.4965$$ $$\Gamma_0(N)$$-optimal
31200.c2 31200bp2 $$[0, -1, 0, -62433, -9673263]$$ $$-420526439488/390971529$$ $$-25022177856000000$$ $$[2]$$ $$196608$$ $$1.8431$$

## Rank

sage: E.rank()

The elliptic curves in class 31200bp have rank $$1$$.

## Complex multiplication

The elliptic curves in class 31200bp do not have complex multiplication.

## Modular form 31200.2.a.bp

sage: E.q_eigenform(10)

$$q - q^{3} - 2 q^{7} + q^{9} - 2 q^{11} + q^{13} - 6 q^{17} + 2 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.