# Properties

 Label 323400.cz Number of curves $4$ Conductor $323400$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("cz1")

sage: E.isogeny_class()

## Elliptic curves in class 323400.cz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
323400.cz1 323400cz4 $$[0, -1, 0, -36476008, -84766747988]$$ $$1425631925916578/270703125$$ $$1019134462500000000000$$ $$$$ $$18874368$$ $$3.0322$$
323400.cz2 323400cz3 $$[0, -1, 0, -15994008, 23846064012]$$ $$120186986927618/4332064275$$ $$16309216956463200000000$$ $$$$ $$18874368$$ $$3.0322$$
323400.cz3 323400cz2 $$[0, -1, 0, -2519008, -1028785988]$$ $$939083699236/300155625$$ $$565008146010000000000$$ $$[2, 2]$$ $$9437184$$ $$2.6856$$
323400.cz4 323400cz1 $$[0, -1, 0, 445492, -109790988]$$ $$20777545136/23059575$$ $$-10851743756700000000$$ $$$$ $$4718592$$ $$2.3391$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 323400.cz have rank $$1$$.

## Complex multiplication

The elliptic curves in class 323400.cz do not have complex multiplication.

## Modular form 323400.2.a.cz

sage: E.q_eigenform(10)

$$q - q^{3} + q^{9} + q^{11} - 2 q^{13} - 2 q^{17} + O(q^{20})$$

## 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. 