# Properties

 Label 405.c Number of curves $2$ Conductor $405$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 405.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405.c1 405b2 $$[0, 0, 1, -108, -412]$$ $$2359296/125$$ $$7381125$$ $$[]$$ $$72$$ $$0.074670$$
405.c2 405b1 $$[0, 0, 1, -18, 29]$$ $$884736/5$$ $$3645$$ $$$$ $$24$$ $$-0.47464$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 405.c have rank $$1$$.

## Complex multiplication

The elliptic curves in class 405.c do not have complex multiplication.

## Modular form405.2.a.c

sage: E.q_eigenform(10)

$$q - 2 q^{4} - q^{5} + 2 q^{7} - 3 q^{11} - 4 q^{13} + 4 q^{16} - 6 q^{17} - q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 