# Properties

 Label 30345m Number of curves 4 Conductor 30345 CM no Rank 0 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("30345.f1")

sage: E.isogeny_class()

## Elliptic curves in class 30345m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
30345.f4 30345m1 [1, 1, 1, 4040, 167912]  73728 $$\Gamma_0(N)$$-optimal
30345.f3 30345m2 [1, 1, 1, -32085, 1800762] [2, 2] 147456
30345.f2 30345m3 [1, 1, 1, -154910, -21879898]  294912
30345.f1 30345m4 [1, 1, 1, -487260, 130706322]  294912

## Rank

sage: E.rank()

The elliptic curves in class 30345m have rank $$0$$.

## Modular form 30345.2.a.f

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 