# Properties

 Label 6840.g Number of curves $4$ Conductor $6840$ CM no Rank $0$ Graph

# Learn more

Show commands: SageMath
sage: E = EllipticCurve("g1")

sage: E.isogeny_class()

## Elliptic curves in class 6840.g

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6840.g1 6840r3 $$[0, 0, 0, -273603, -55084498]$$ $$3034301922374404/1425$$ $$1063756800$$ $$[2]$$ $$16384$$ $$1.5058$$
6840.g2 6840r4 $$[0, 0, 0, -20523, -491722]$$ $$1280615525284/601171875$$ $$448772400000000$$ $$[2]$$ $$16384$$ $$1.5058$$
6840.g3 6840r2 $$[0, 0, 0, -17103, -860398]$$ $$2964647793616/2030625$$ $$378963360000$$ $$[2, 2]$$ $$8192$$ $$1.1592$$
6840.g4 6840r1 $$[0, 0, 0, -858, -18907]$$ $$-5988775936/9774075$$ $$-114004810800$$ $$[4]$$ $$4096$$ $$0.81261$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 6840.g have rank $$0$$.

## Complex multiplication

The elliptic curves in class 6840.g do not have complex multiplication.

## Modular form6840.2.a.g

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.