# Properties

 Label 3450.p Number of curves $4$ Conductor $3450$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("p1")

sage: E.isogeny_class()

## Elliptic curves in class 3450.p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3450.p1 3450o3 $$[1, 1, 1, -266088, 52603281]$$ $$133345896593725369/340006815000$$ $$5312606484375000$$ $$$$ $$46080$$ $$1.8939$$
3450.p2 3450o2 $$[1, 1, 1, -23088, 115281]$$ $$87109155423289/49979073600$$ $$780923025000000$$ $$[2, 2]$$ $$23040$$ $$1.5473$$
3450.p3 3450o1 $$[1, 1, 1, -15088, -716719]$$ $$24310870577209/114462720$$ $$1788480000000$$ $$$$ $$11520$$ $$1.2007$$ $$\Gamma_0(N)$$-optimal
3450.p4 3450o4 $$[1, 1, 1, 91912, 1035281]$$ $$5495662324535111/3207841648920$$ $$-50122525764375000$$ $$$$ $$46080$$ $$1.8939$$

## Rank

sage: E.rank()

The elliptic curves in class 3450.p have rank $$0$$.

## Complex multiplication

The elliptic curves in class 3450.p do not have complex multiplication.

## Modular form3450.2.a.p

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{8} + q^{9} - 4q^{11} - q^{12} + 6q^{13} + q^{16} + 6q^{17} + q^{18} + 4q^{19} + O(q^{20})$$ ## 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 LMFDB numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 