# Properties

 Label 312.c Number of curves $2$ Conductor $312$ CM no Rank $0$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("c1")

sage: E.isogeny_class()

## Elliptic curves in class 312.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
312.c1 312e1 $$[0, -1, 0, -651, 6228]$$ $$1909913257984/129730653$$ $$2075690448$$ $$$$ $$240$$ $$0.53639$$ $$\Gamma_0(N)$$-optimal
312.c2 312e2 $$[0, -1, 0, 564, 25668]$$ $$77366117936/1172914587$$ $$-300266134272$$ $$$$ $$480$$ $$0.88296$$

## Rank

sage: E.rank()

The elliptic curves in class 312.c have rank $$0$$.

## Complex multiplication

The elliptic curves in class 312.c do not have complex multiplication.

## Modular form312.2.a.c

sage: E.q_eigenform(10)

$$q - q^{3} + 4 q^{5} + q^{9} - 2 q^{11} - q^{13} - 4 q^{15} + 2 q^{17} + 8 q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 