# Properties

 Label 317900e Number of curves $4$ Conductor $317900$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("e1")

sage: E.isogeny_class()

## Elliptic curves in class 317900e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
317900.e4 317900e1 $$[0, 1, 0, -327533, 70560688]$$ $$643956736/15125$$ $$91270182781250000$$ $$$$ $$4478976$$ $$2.0399$$ $$\Gamma_0(N)$$-optimal
317900.e3 317900e2 $$[0, 1, 0, -724908, -134484812]$$ $$436334416/171875$$ $$16594578687500000000$$ $$$$ $$8957952$$ $$2.3865$$
317900.e2 317900e3 $$[0, 1, 0, -3217533, -2194476812]$$ $$610462990336/8857805$$ $$53451469844011250000$$ $$$$ $$13436928$$ $$2.5892$$
317900.e1 317900e4 $$[0, 1, 0, -51299908, -141441034812]$$ $$154639330142416/33275$$ $$3212710433900000000$$ $$$$ $$26873856$$ $$2.9358$$

## Rank

sage: E.rank()

The elliptic curves in class 317900e have rank $$1$$.

## Complex multiplication

The elliptic curves in class 317900e do not have complex multiplication.

## Modular form 317900.2.a.e

sage: E.q_eigenform(10)

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

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 