# Properties

 Label 72450q Number of curves $2$ Conductor $72450$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("q1")

sage: E.isogeny_class()

## Elliptic curves in class 72450q

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
72450.ca2 72450q1 $$[1, -1, 0, -1527, -680419]$$ $$-160103007/81288256$$ $$-199999592856000$$ $$$$ $$221184$$ $$1.4232$$ $$\Gamma_0(N)$$-optimal
72450.ca1 72450q2 $$[1, -1, 0, -125727, -16950619]$$ $$89332607016927/1060723384$$ $$2609777295909000$$ $$$$ $$442368$$ $$1.7698$$

## Rank

sage: E.rank()

The elliptic curves in class 72450q have rank $$1$$.

## Complex multiplication

The elliptic curves in class 72450q do not have complex multiplication.

## Modular form 72450.2.a.q

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + q^{7} - q^{8} + 2q^{11} + 2q^{13} - q^{14} + q^{16} + 2q^{17} - 4q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 