# Properties

 Label 310464.dz Number of curves $4$ Conductor $310464$ CM no Rank $0$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("dz1")

sage: E.isogeny_class()

## Elliptic curves in class 310464.dz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
310464.dz1 310464dz3 $$[0, 0, 0, -1129369836, -10358298308176]$$ $$14171198121996897746/4077720290568771$$ $$45839843569837850284168445952$$ $$[2]$$ $$283115520$$ $$4.2060$$
310464.dz2 310464dz2 $$[0, 0, 0, -1035454476, -12823050582160]$$ $$21843440425782779332/3100814593569$$ $$17428961010031308036440064$$ $$[2, 2]$$ $$141557760$$ $$3.8594$$
310464.dz3 310464dz1 $$[0, 0, 0, -1035419196, -12823968186736]$$ $$87364831012240243408/1760913$$ $$2474421082988101632$$ $$[2]$$ $$70778880$$ $$3.5128$$ $$\Gamma_0(N)$$-optimal
310464.dz4 310464dz4 $$[0, 0, 0, -942103596, -15229076163280]$$ $$-8226100326647904626/4152140742401883$$ $$-46676443833548108989181853696$$ $$[2]$$ $$283115520$$ $$4.2060$$

## Rank

sage: E.rank()

The elliptic curves in class 310464.dz have rank $$0$$.

## Complex multiplication

The elliptic curves in class 310464.dz do not have complex multiplication.

## Modular form 310464.2.a.dz

sage: E.q_eigenform(10)

$$q - 2 q^{5} + q^{11} - 6 q^{13} - 2 q^{17} - 8 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.