# Properties

 Label 6930.p Number of curves $4$ Conductor $6930$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("p1")

sage: E.isogeny_class()

## Elliptic curves in class 6930.p

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6930.p1 6930q3 $$[1, -1, 0, -218349, 39325905]$$ $$1579250141304807889/41926500$$ $$30564418500$$ $$$$ $$41472$$ $$1.5260$$
6930.p2 6930q4 $$[1, -1, 0, -218079, 39427803]$$ $$-1573398910560073969/8138108343750$$ $$-5932680982593750$$ $$$$ $$82944$$ $$1.8726$$
6930.p3 6930q1 $$[1, -1, 0, -2889, 46413]$$ $$3658671062929/880165440$$ $$641640605760$$ $$$$ $$13824$$ $$0.97671$$ $$\Gamma_0(N)$$-optimal
6930.p4 6930q2 $$[1, -1, 0, 6831, 285525]$$ $$48351870250991/76871856600$$ $$-56039583461400$$ $$$$ $$27648$$ $$1.3233$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form6930.2.a.p

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 