# Properties

 Label 30345.t Number of curves 4 Conductor 30345 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("30345.t1")

sage: E.isogeny_class()

## Elliptic curves in class 30345.t

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
30345.t1 30345b4 [1, 1, 0, -32518, -2270453]  81920
30345.t2 30345b2 [1, 1, 0, -2173, -30992] [2, 2] 40960
30345.t3 30345b1 [1, 1, 0, -728, 6867]  20480 $$\Gamma_0(N)$$-optimal
30345.t4 30345b3 [1, 1, 0, 5052, -185607]  81920

## Rank

sage: E.rank()

The elliptic curves in class 30345.t have rank $$1$$.

## Modular form 30345.2.a.t

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} - q^{5} - q^{6} - q^{7} - 3q^{8} + q^{9} - q^{10} + q^{12} - 6q^{13} - q^{14} + q^{15} - q^{16} + q^{18} - 8q^{19} + O(q^{20})$$

## 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. 