# Properties

 Label 6930.z Number of curves $8$ Conductor $6930$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("z1")

sage: E.isogeny_class()

## Elliptic curves in class 6930.z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6930.z1 6930ba7 $$[1, -1, 1, -2307848, -1265537149]$$ $$1864737106103260904761/129177711985836360$$ $$94170552037674706440$$ $$[6]$$ $$221184$$ $$2.5806$$
6930.z2 6930ba4 $$[1, -1, 1, -2268023, -1314110419]$$ $$1769857772964702379561/691787250$$ $$504312905250$$ $$[2]$$ $$73728$$ $$2.0313$$
6930.z3 6930ba6 $$[1, -1, 1, -455648, 94718531]$$ $$14351050585434661561/3001282273281600$$ $$2187934777222286400$$ $$[2, 6]$$ $$110592$$ $$2.2340$$
6930.z4 6930ba3 $$[1, -1, 1, -429728, 108528707]$$ $$12038605770121350841/757333463040$$ $$552096094556160$$ $$[6]$$ $$55296$$ $$1.8874$$
6930.z5 6930ba2 $$[1, -1, 1, -141773, -20499919]$$ $$432288716775559561/270140062500$$ $$196932105562500$$ $$[2, 2]$$ $$36864$$ $$1.6847$$
6930.z6 6930ba5 $$[1, -1, 1, -115043, -28486843]$$ $$-230979395175477481/348191894531250$$ $$-253831891113281250$$ $$[2]$$ $$73728$$ $$2.0313$$
6930.z7 6930ba1 $$[1, -1, 1, -10553, -187063]$$ $$178272935636041/81841914000$$ $$59662755306000$$ $$[2]$$ $$18432$$ $$1.3381$$ $$\Gamma_0(N)$$-optimal
6930.z8 6930ba8 $$[1, -1, 1, 981832, 570811907]$$ $$143584693754978072519/276341298967965000$$ $$-201452806947646485000$$ $$[6]$$ $$221184$$ $$2.5806$$

## Rank

sage: E.rank()

The elliptic curves in class 6930.z have rank $$1$$.

## Complex multiplication

The elliptic curves in class 6930.z do not have complex multiplication.

## Modular form6930.2.a.z

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - q^{5} + q^{7} + q^{8} - q^{10} + q^{11} + 2 q^{13} + q^{14} + q^{16} - 6 q^{17} - 4 q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.