# Properties

 Label 51600.bz Number of curves $4$ Conductor $51600$ CM no Rank $0$ Graph # Learn more

Show commands for: SageMath
sage: E = EllipticCurve("bz1")

sage: E.isogeny_class()

## Elliptic curves in class 51600.bz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
51600.bz1 51600x4 $$[0, 1, 0, -229408, -42368812]$$ $$41725476313778/17415$$ $$557280000000$$ $$$$ $$294912$$ $$1.5984$$
51600.bz2 51600x2 $$[0, 1, 0, -14408, -658812]$$ $$20674973956/416025$$ $$6656400000000$$ $$[2, 2]$$ $$147456$$ $$1.2518$$
51600.bz3 51600x1 $$[0, 1, 0, -1908, 16188]$$ $$192143824/80625$$ $$322500000000$$ $$$$ $$73728$$ $$0.90525$$ $$\Gamma_0(N)$$-optimal
51600.bz4 51600x3 $$[0, 1, 0, 592, -1948812]$$ $$715822/51282015$$ $$-1641024480000000$$ $$$$ $$294912$$ $$1.5984$$

## Rank

sage: E.rank()

The elliptic curves in class 51600.bz have rank $$0$$.

## Complex multiplication

The elliptic curves in class 51600.bz do not have complex multiplication.

## Modular form 51600.2.a.bz

sage: E.q_eigenform(10)

$$q + q^{3} - 4q^{7} + q^{9} - 4q^{11} + 2q^{13} - 6q^{17} + 8q^{19} + O(q^{20})$$ ## 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. 