# Properties

 Label 3450.j Number of curves $4$ Conductor $3450$ CM no Rank $1$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("j1")

sage: E.isogeny_class()

## Elliptic curves in class 3450.j

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3450.j1 3450j3 $$[1, 0, 1, -34560276, -78204168302]$$ $$292169767125103365085489/72534787200$$ $$1133356050000000$$ $$[2]$$ $$172032$$ $$2.7043$$
3450.j2 3450j4 $$[1, 0, 1, -2528276, -777224302]$$ $$114387056741228939569/49503729150000000$$ $$773495767968750000000$$ $$[2]$$ $$172032$$ $$2.7043$$
3450.j3 3450j2 $$[1, 0, 1, -2160276, -1221768302]$$ $$71356102305927901489/35540674560000$$ $$555323040000000000$$ $$[2, 2]$$ $$86016$$ $$2.3578$$
3450.j4 3450j1 $$[1, 0, 1, -112276, -25736302]$$ $$-10017490085065009/12502381363200$$ $$-195349708800000000$$ $$[2]$$ $$43008$$ $$2.0112$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 3450.j have rank $$1$$.

## Complex multiplication

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

## Modular form3450.2.a.j

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} - q^{6} - q^{8} + q^{9} + q^{12} - 6q^{13} + q^{16} - 2q^{17} - q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.