# Properties

 Label 7056.k Number of curves $6$ Conductor $7056$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("k1")

sage: E.isogeny_class()

## Elliptic curves in class 7056.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
7056.k1 7056by3 $$[0, 0, 0, -9483411, -11240741806]$$ $$268498407453697/252$$ $$88527103967232$$ $$[2]$$ $$147456$$ $$2.4045$$
7056.k2 7056by5 $$[0, 0, 0, -6449331, 6243532274]$$ $$84448510979617/933897762$$ $$328076445521187643392$$ $$[2]$$ $$294912$$ $$2.7511$$
7056.k3 7056by4 $$[0, 0, 0, -733971, -85657390]$$ $$124475734657/63011844$$ $$22135936765694459904$$ $$[2, 2]$$ $$147456$$ $$2.4045$$
7056.k4 7056by2 $$[0, 0, 0, -592851, -175550830]$$ $$65597103937/63504$$ $$22308830199742464$$ $$[2, 2]$$ $$73728$$ $$2.0579$$
7056.k5 7056by1 $$[0, 0, 0, -28371, -4061806]$$ $$-7189057/16128$$ $$-5665734653902848$$ $$[2]$$ $$36864$$ $$1.7114$$ $$\Gamma_0(N)$$-optimal
7056.k6 7056by6 $$[0, 0, 0, 2723469, -661666894]$$ $$6359387729183/4218578658$$ $$-1481978378772666851328$$ $$[2]$$ $$294912$$ $$2.7511$$

## Rank

sage: E.rank()

The elliptic curves in class 7056.k have rank $$0$$.

## Complex multiplication

The elliptic curves in class 7056.k do not have complex multiplication.

## Modular form7056.2.a.k

sage: E.q_eigenform(10)

$$q - 2q^{5} - 4q^{11} - 6q^{13} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.