# Properties

 Label 30345.c Number of curves 6 Conductor 30345 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("30345.c1")

sage: E.isogeny_class()

## Elliptic curves in class 30345.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
30345.c1 30345c6 [1, 1, 1, -20732866, -36274177462]  2359296
30345.c2 30345c4 [1, 1, 1, -1767241, -118109962] [2, 2] 1179648
30345.c3 30345c2 [1, 1, 1, -1129996, 459743804] [2, 2] 589824
30345.c4 30345c1 [1, 1, 1, -1128551, 460985348]  294912 $$\Gamma_0(N)$$-optimal
30345.c5 30345c3 [1, 1, 1, -515871, 958167654]  1179648
30345.c6 30345c5 [1, 1, 1, 7002464, -931938586]  2359296

## Rank

sage: E.rank()

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

## Modular form 30345.2.a.c

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} - q^{4} - q^{5} + q^{6} - q^{7} + 3q^{8} + q^{9} + q^{10} + 4q^{11} + q^{12} - 2q^{13} + q^{14} + q^{15} - q^{16} - q^{18} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 