# Properties

 Label 40344.c Number of curves $6$ Conductor $40344$ CM no Rank $1$ Graph # Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("c1")

sage: E.isogeny_class()

## Elliptic curves in class 40344.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
40344.c1 40344g6 $$[0, 1, 0, -646064, -200091360]$$ $$3065617154/9$$ $$87553921370112$$ $$$$ $$276480$$ $$1.9046$$
40344.c2 40344g4 $$[0, 1, 0, -108144, 13651152]$$ $$28756228/3$$ $$14592320228352$$ $$$$ $$138240$$ $$1.5580$$
40344.c3 40344g3 $$[0, 1, 0, -40904, -3051264]$$ $$1556068/81$$ $$393992646165504$$ $$[2, 2]$$ $$138240$$ $$1.5580$$
40344.c4 40344g2 $$[0, 1, 0, -7284, 176256]$$ $$35152/9$$ $$10944240171264$$ $$[2, 2]$$ $$69120$$ $$1.2114$$
40344.c5 40344g1 $$[0, 1, 0, 1121, 18242]$$ $$2048/3$$ $$-228005003568$$ $$$$ $$34560$$ $$0.86486$$ $$\Gamma_0(N)$$-optimal
40344.c6 40344g5 $$[0, 1, 0, 26336, -12034528]$$ $$207646/6561$$ $$-63826808678811648$$ $$$$ $$276480$$ $$1.9046$$

## Rank

sage: E.rank()

The elliptic curves in class 40344.c have rank $$1$$.

## Complex multiplication

The elliptic curves in class 40344.c do not have complex multiplication.

## Modular form 40344.2.a.c

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{5} + q^{9} - 4q^{11} + 2q^{13} - 2q^{15} - 2q^{17} + 4q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 