| $\GL_2(\Z/14\Z)$-generators: |
$\begin{bmatrix}7&9\\2&7\end{bmatrix}$, $\begin{bmatrix}10&3\\11&9\end{bmatrix}$ |
| $\GL_2(\Z/14\Z)$-subgroup: |
$C_3\times D_{42}$ |
| Contains $-I$: |
yes |
| Quadratic refinements: |
14.96.2-14.g.1.1, 14.96.2-14.g.1.2, 28.96.2-14.g.1.1, 28.96.2-14.g.1.2, 42.96.2-14.g.1.1, 42.96.2-14.g.1.2, 56.96.2-14.g.1.1, 56.96.2-14.g.1.2, 56.96.2-14.g.1.3, 56.96.2-14.g.1.4, 70.96.2-14.g.1.1, 70.96.2-14.g.1.2, 84.96.2-14.g.1.1, 84.96.2-14.g.1.2, 140.96.2-14.g.1.1, 140.96.2-14.g.1.2, 154.96.2-14.g.1.1, 154.96.2-14.g.1.2, 168.96.2-14.g.1.1, 168.96.2-14.g.1.2, 168.96.2-14.g.1.3, 168.96.2-14.g.1.4, 182.96.2-14.g.1.1, 182.96.2-14.g.1.2, 210.96.2-14.g.1.1, 210.96.2-14.g.1.2, 238.96.2-14.g.1.1, 238.96.2-14.g.1.2, 266.96.2-14.g.1.1, 266.96.2-14.g.1.2, 280.96.2-14.g.1.1, 280.96.2-14.g.1.2, 280.96.2-14.g.1.3, 280.96.2-14.g.1.4, 308.96.2-14.g.1.1, 308.96.2-14.g.1.2, 322.96.2-14.g.1.1, 322.96.2-14.g.1.2 |
| Cyclic 14-isogeny field degree: |
$3$ |
| Cyclic 14-torsion field degree: |
$18$ |
| Full 14-torsion field degree: |
$252$ |
Embedded model Embedded model in $\mathbb{P}^{3}$
| $ 0 $ | $=$ | $ 2 x z w + x w^{2} - y z w + 3 y w^{2} $ |
| $=$ | $2 x z^{2} + x z w - y z^{2} + 3 y z w$ |
| $=$ | $2 x y z + x y w - y^{2} z + 3 y^{2} w$ |
| $=$ | $2 x^{2} z + x^{2} w - x y z + 3 x y w$ |
| $=$ | $\cdots$ |
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Singular plane model Singular plane model
| $ 0 $ | $=$ | $ 4 x^{5} - x^{3} y^{2} - 11 x^{3} z^{2} + 2 x^{2} y^{2} z - 5 x^{2} z^{3} + x y^{2} z^{2} + 2 x z^{4} + \cdots + z^{5} $ |
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Weierstrass model Weierstrass model
| $ y^{2} $ | $=$ | $ x^{6} - 3x^{5} - x^{4} + 7x^{3} - x^{2} - 3x + 1 $ |
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This modular curve has 1 rational CM point but no rational cusps or other known rational points. The following are the known rational points on this modular curve (one row per $j$-invariant).
| Elliptic curve |
CM |
$j$-invariant |
$j$-height | Plane model | Weierstrass model | Embedded model |
| 49.a2 |
$-7$ | $-3375$ |
$= -1 \cdot 3^{3} \cdot 5^{3}$ | $8.124$ | $(0:1:1)$, $(1:3:1)$, $(1:-3:1)$, $(0:-1:1)$, $(-1/2:1:0)$, $(1/2:1:0)$ | $(1:1:0)$, $(0:1:1)$, $(0:-1:1)$, $(1:-1:0)$, $(1:1:1)$, $(1:-1:1)$ | $(-3:1:0:1)$, $(-2:3:1:1)$, $(2:-3:1:1)$, $(3:-1:0:1)$, $(-1:-2:1:0)$, $(1:2:1:0)$ |
Maps between models of this curve
Birational map from embedded model to plane model:
| $\displaystyle X$ |
$=$ |
$\displaystyle z$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle y$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle w$ |
Birational map from embedded model to Weierstrass model:
| $\displaystyle X$ |
$=$ |
$\displaystyle -z^{2}+\frac{1}{2}zw+\frac{1}{2}w^{2}$ |
| $\displaystyle Y$ |
$=$ |
$\displaystyle \frac{1}{2}yz^{5}-\frac{1}{2}yz^{4}w-\frac{11}{8}yz^{3}w^{2}-\frac{1}{4}yz^{2}w^{3}+\frac{3}{8}yzw^{4}+\frac{1}{8}yw^{5}$ |
| $\displaystyle Z$ |
$=$ |
$\displaystyle -z^{2}-\frac{1}{2}zw$ |
Maps to other modular curves
$j$-invariant map
of degree 48 from the embedded model of this modular curve to the modular curve
$X(1)$
:
| $\displaystyle j$ |
$=$ |
$\displaystyle \frac{77861x^{2}y^{8}-4872511x^{2}y^{6}w^{2}-10907008x^{2}y^{4}w^{4}-36199093x^{2}y^{2}w^{6}-595491127x^{2}w^{8}+100842xy^{9}-6668067xy^{7}w^{2}-4500552xy^{5}w^{4}+152224772xy^{3}w^{6}+3165561315xyw^{8}+12348y^{10}-1124403y^{8}w^{2}+6182673y^{6}w^{4}+197095836y^{4}w^{6}+4169687326y^{2}w^{8}-84224747z^{10}-521984694z^{9}w-64881341z^{8}w^{2}+3042644335z^{7}w^{3}+4515844226z^{6}w^{4}+208775624z^{5}w^{5}-6693229757z^{4}w^{6}-8522985076z^{3}w^{7}-21626086224z^{2}w^{8}-13765983899zw^{9}+11374814552w^{10}}{(5z+4w)(z^{3}-2z^{2}w-zw^{2}+w^{3})^{2}(z^{3}-z^{2}w-2zw^{2}+w^{3})}$ |
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Cover information
Click on a modular curve in the diagram to see information about it.
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This modular curve minimally covers the modular curves listed below.
This modular curve is minimally covered by the modular curves in the database listed below.
