# Properties

 Label 397800.cb Number of curves $4$ Conductor $397800$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("cb1")

sage: E.isogeny_class()

## Elliptic curves in class 397800.cb

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
397800.cb1 397800cb3 $$[0, 0, 0, -14061675, 20295521750]$$ $$26362547147244676/244298925$$ $$2849502661200000000$$ $$$$ $$14155776$$ $$2.7048$$
397800.cb2 397800cb2 $$[0, 0, 0, -899175, 301684250]$$ $$27572037674704/2472575625$$ $$7210030522500000000$$ $$[2, 2]$$ $$7077888$$ $$2.3582$$
397800.cb3 397800cb1 $$[0, 0, 0, -196050, -28081375]$$ $$4572531595264/776953125$$ $$141599707031250000$$ $$$$ $$3538944$$ $$2.0117$$ $$\Gamma_0(N)$$-optimal
397800.cb4 397800cb4 $$[0, 0, 0, 1013325, 1412846750]$$ $$9865576607324/79640206425$$ $$-928923367741200000000$$ $$$$ $$14155776$$ $$2.7048$$

## Rank

sage: E.rank()

The elliptic curves in class 397800.cb have rank $$1$$.

## Complex multiplication

The elliptic curves in class 397800.cb do not have complex multiplication.

## Modular form 397800.2.a.cb

sage: E.q_eigenform(10)

$$q - 4q^{11} - q^{13} + q^{17} - 8q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 