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Properties

Label 14.24.1.a.1
Level $14$
Index $24$
Genus $1$
Analytic rank $0$
Cusps $4$
$\Q$-cusps $4$

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Invariants

Level: $14$ $\SL_2$-level: $14$ Newform level: $14$
Index: $24$ $\PSL_2$-index:$24$
Genus: $1 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 4 }{2}$
Cusps: $4$ (all of which are rational) Cusp widths $1\cdot2\cdot7\cdot14$ Cusp orbits $1^{4}$
Elliptic points: $0$ of order $2$ and $0$ of order $3$
Analytic rank: $0$
$\Q$-gonality: $2$
$\overline{\Q}$-gonality: $2$
Rational cusps: $4$
Rational CM points: yes $\quad(D =$ $-7,-28$)

Other labels

Cummins and Pauli (CP) label: 14C1
Rouse, Sutherland, and Zureick-Brown (RSZB) label: 14.24.1.1

Level structure

$\GL_2(\Z/14\Z)$-generators: $\begin{bmatrix}1&3\\0&13\end{bmatrix}$, $\begin{bmatrix}5&2\\0&3\end{bmatrix}$, $\begin{bmatrix}5&5\\0&5\end{bmatrix}$
$\GL_2(\Z/14\Z)$-subgroup: $C_{14}:C_6^2$
Contains $-I$: yes
Quadratic refinements: 14.48.1-14.a.1.1, 14.48.1-14.a.1.2, 14.48.1-14.a.1.3, 14.48.1-14.a.1.4, 28.48.1-14.a.1.1, 28.48.1-14.a.1.2, 28.48.1-14.a.1.3, 28.48.1-14.a.1.4, 28.48.1-14.a.1.5, 28.48.1-14.a.1.6, 28.48.1-14.a.1.7, 28.48.1-14.a.1.8, 28.48.1-14.a.1.9, 28.48.1-14.a.1.10, 28.48.1-14.a.1.11, 28.48.1-14.a.1.12, 42.48.1-14.a.1.1, 42.48.1-14.a.1.2, 42.48.1-14.a.1.3, 42.48.1-14.a.1.4, 56.48.1-14.a.1.1, 56.48.1-14.a.1.2, 56.48.1-14.a.1.3, 56.48.1-14.a.1.4, 56.48.1-14.a.1.5, 56.48.1-14.a.1.6, 56.48.1-14.a.1.7, 56.48.1-14.a.1.8, 56.48.1-14.a.1.9, 56.48.1-14.a.1.10, 56.48.1-14.a.1.11, 56.48.1-14.a.1.12, 56.48.1-14.a.1.13, 56.48.1-14.a.1.14, 56.48.1-14.a.1.15, 56.48.1-14.a.1.16, 70.48.1-14.a.1.1, 70.48.1-14.a.1.2, 70.48.1-14.a.1.3, 70.48.1-14.a.1.4, 84.48.1-14.a.1.1, 84.48.1-14.a.1.2, 84.48.1-14.a.1.3, 84.48.1-14.a.1.4, 84.48.1-14.a.1.5, 84.48.1-14.a.1.6, 84.48.1-14.a.1.7, 84.48.1-14.a.1.8, 84.48.1-14.a.1.9, 84.48.1-14.a.1.10, 84.48.1-14.a.1.11, 84.48.1-14.a.1.12, 140.48.1-14.a.1.1, 140.48.1-14.a.1.2, 140.48.1-14.a.1.3, 140.48.1-14.a.1.4, 140.48.1-14.a.1.5, 140.48.1-14.a.1.6, 140.48.1-14.a.1.7, 140.48.1-14.a.1.8, 140.48.1-14.a.1.9, 140.48.1-14.a.1.10, 140.48.1-14.a.1.11, 140.48.1-14.a.1.12, 154.48.1-14.a.1.1, 154.48.1-14.a.1.2, 154.48.1-14.a.1.3, 154.48.1-14.a.1.4, 168.48.1-14.a.1.1, 168.48.1-14.a.1.2, 168.48.1-14.a.1.3, 168.48.1-14.a.1.4, 168.48.1-14.a.1.5, 168.48.1-14.a.1.6, 168.48.1-14.a.1.7, 168.48.1-14.a.1.8, 168.48.1-14.a.1.9, 168.48.1-14.a.1.10, 168.48.1-14.a.1.11, 168.48.1-14.a.1.12, 168.48.1-14.a.1.13, 168.48.1-14.a.1.14, 168.48.1-14.a.1.15, 168.48.1-14.a.1.16, 182.48.1-14.a.1.1, 182.48.1-14.a.1.2, 182.48.1-14.a.1.3, 182.48.1-14.a.1.4, 210.48.1-14.a.1.1, 210.48.1-14.a.1.2, 210.48.1-14.a.1.3, 210.48.1-14.a.1.4, 238.48.1-14.a.1.1, 238.48.1-14.a.1.2, 238.48.1-14.a.1.3, 238.48.1-14.a.1.4, 266.48.1-14.a.1.1, 266.48.1-14.a.1.2, 266.48.1-14.a.1.3, 266.48.1-14.a.1.4, 280.48.1-14.a.1.1, 280.48.1-14.a.1.2, 280.48.1-14.a.1.3, 280.48.1-14.a.1.4, 280.48.1-14.a.1.5, 280.48.1-14.a.1.6, 280.48.1-14.a.1.7, 280.48.1-14.a.1.8, 280.48.1-14.a.1.9, 280.48.1-14.a.1.10, 280.48.1-14.a.1.11, 280.48.1-14.a.1.12, 280.48.1-14.a.1.13, 280.48.1-14.a.1.14, 280.48.1-14.a.1.15, 280.48.1-14.a.1.16, 308.48.1-14.a.1.1, 308.48.1-14.a.1.2, 308.48.1-14.a.1.3, 308.48.1-14.a.1.4, 308.48.1-14.a.1.5, 308.48.1-14.a.1.6, 308.48.1-14.a.1.7, 308.48.1-14.a.1.8, 308.48.1-14.a.1.9, 308.48.1-14.a.1.10, 308.48.1-14.a.1.11, 308.48.1-14.a.1.12, 322.48.1-14.a.1.1, 322.48.1-14.a.1.2, 322.48.1-14.a.1.3, 322.48.1-14.a.1.4
Cyclic 14-isogeny field degree: $1$
Cyclic 14-torsion field degree: $6$
Full 14-torsion field degree: $504$

Jacobian

Conductor: $2\cdot7$
Simple: yes
Squarefree: yes
Decomposition: $1$
Newforms: 14.2.a.a

Models

Weierstrass model Weierstrass model

$ y^{2} + \left(x + 1\right) y $ $=$ $ x^{3} + 4x - 6 $
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Rational points

This modular curve has 4 rational cusps and 2 rational CM points, but no 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 $j$-heightWeierstrass model
no$\infty$ $0.000$$(1:-1:1)$, $(2:-5:1)$, $(0:1:0)$, $(9:23:1)$
49.a2 $-7$$-3375$ $= -1 \cdot 3^{3} \cdot 5^{3}$$8.124$$(9:-33:1)$
49.a1 $-28$$16581375$ $= 3^{3} \cdot 5^{3} \cdot 17^{3}$$16.624$$(2:2:1)$

Maps to other modular curves

$j$-invariant map of degree 24 from the Weierstrass model of this modular curve to the modular curve $X(1)$ :

$\displaystyle j$ $=$ $\displaystyle -\frac{34x^{2}y^{7}-182x^{2}y^{6}z-11430x^{2}y^{5}z^{2}-154440x^{2}y^{4}z^{3}+672354x^{2}y^{3}z^{4}+10402236x^{2}y^{2}z^{5}-14609565x^{2}yz^{6}-121236750x^{2}z^{7}-xy^{8}-449xy^{7}z+810xy^{6}z^{2}+50814xy^{5}z^{3}+415863xy^{4}z^{4}+1872126xy^{3}z^{5}-30769146xy^{2}z^{6}-123710868xyz^{7}+342481419xz^{8}-y^{9}+6y^{8}z+3075y^{7}z^{2}+23652y^{6}z^{3}-233127y^{5}z^{4}-1986390y^{4}z^{5}-7202520y^{3}z^{6}+7585785y^{2}z^{7}+238368771yz^{8}+22654080z^{9}}{z^{2}(3x^{2}y^{5}+192x^{2}y^{4}z-1472x^{2}y^{3}z^{2}+3844x^{2}y^{2}z^{3}-19398x^{2}yz^{4}+36573x^{2}z^{5}+xy^{6}-74xy^{5}z-490xy^{4}z^{2}-1262xy^{3}z^{3}-13320xy^{2}z^{4}-18988xyz^{5}-122459xz^{6}-24y^{6}z+430y^{5}z^{2}+1547y^{4}z^{3}+10953y^{3}z^{4}+29698y^{2}z^{5}+57236yz^{6}+91866z^{7})}$

Modular covers

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Cover information

Click on a modular curve in the diagram to see information about it.
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The following modular covers realize this modular curve as a fiber product over $X(1)$.

Factor curve Level Index Degree Genus Rank Kernel decomposition
$X_0(2)$ $2$ $8$ $8$ $0$ $0$ full Jacobian
$X_0(7)$ $7$ $3$ $3$ $0$ $0$ full Jacobian

This modular curve minimally covers the modular curves listed below.

Covered curve Level Index Degree Genus Rank Kernel decomposition
$X_0(2)$ $2$ $8$ $8$ $0$ $0$ full Jacobian
$X_0(7)$ $7$ $3$ $3$ $0$ $0$ full Jacobian

This modular curve is minimally covered by the modular curves in the database listed below.

Covering curve Level Index Degree Genus Rank Kernel decomposition
14.48.2.a.1 $14$ $2$ $2$ $2$ $0$ $1$
14.48.2.f.1 $14$ $2$ $2$ $2$ $0$ $1$
$X_{\pm1}(14)$ $14$ $3$ $3$ $1$ $0$ dimension zero
14.72.1.a.2 $14$ $3$ $3$ $1$ $0$ dimension zero
14.72.1.b.1 $14$ $3$ $3$ $1$ $0$ dimension zero
14.168.7.a.1 $14$ $7$ $7$ $7$ $0$ $1^{4}\cdot2$
28.48.2.a.1 $28$ $2$ $2$ $2$ $0$ $1$
28.48.2.b.1 $28$ $2$ $2$ $2$ $0$ $1$
$X_0(28)$ $28$ $2$ $2$ $2$ $0$ $1$
28.48.2.g.1 $28$ $2$ $2$ $2$ $1$ $1$
28.48.2.h.1 $28$ $2$ $2$ $2$ $1$ $1$
28.48.2.i.1 $28$ $2$ $2$ $2$ $0$ $1$
28.48.3.a.1 $28$ $2$ $2$ $3$ $1$ $1^{2}$
28.48.3.b.1 $28$ $2$ $2$ $3$ $0$ $1^{2}$
28.48.3.c.1 $28$ $2$ $2$ $3$ $0$ $1^{2}$
28.48.3.d.1 $28$ $2$ $2$ $3$ $1$ $1^{2}$
42.48.2.c.1 $42$ $2$ $2$ $2$ $0$ $1$
42.48.2.f.1 $42$ $2$ $2$ $2$ $0$ $1$
42.72.5.a.1 $42$ $3$ $3$ $5$ $0$ $2^{2}$
$X_0(42)$ $42$ $4$ $4$ $5$ $0$ $1^{4}$
56.48.2.b.1 $56$ $2$ $2$ $2$ $0$ $1$
56.48.2.c.1 $56$ $2$ $2$ $2$ $0$ $1$
56.48.2.d.1 $56$ $2$ $2$ $2$ $0$ $1$
56.48.2.e.1 $56$ $2$ $2$ $2$ $0$ $1$
56.48.2.l.1 $56$ $2$ $2$ $2$ $1$ $1$
56.48.2.m.1 $56$ $2$ $2$ $2$ $0$ $1$
56.48.2.n.1 $56$ $2$ $2$ $2$ $0$ $1$
56.48.2.o.1 $56$ $2$ $2$ $2$ $1$ $1$
56.48.3.a.1 $56$ $2$ $2$ $3$ $2$ $1^{2}$
56.48.3.b.1 $56$ $2$ $2$ $3$ $0$ $1^{2}$
56.48.3.c.1 $56$ $2$ $2$ $3$ $1$ $1^{2}$
56.48.3.d.1 $56$ $2$ $2$ $3$ $1$ $1^{2}$
70.48.2.c.1 $70$ $2$ $2$ $2$ $0$ $1$
70.48.2.e.1 $70$ $2$ $2$ $2$ $1$ $1$
70.120.9.a.1 $70$ $5$ $5$ $9$ $1$ $1^{4}\cdot2^{2}$
$X_0(70)$ $70$ $6$ $6$ $9$ $0$ $1^{4}\cdot2^{2}$
70.240.17.a.1 $70$ $10$ $10$ $17$ $3$ $1^{8}\cdot2^{4}$
84.48.2.i.1 $84$ $2$ $2$ $2$ $?$ not computed
84.48.2.j.1 $84$ $2$ $2$ $2$ $?$ not computed
84.48.2.k.1 $84$ $2$ $2$ $2$ $?$ not computed
84.48.2.q.1 $84$ $2$ $2$ $2$ $?$ not computed
84.48.2.r.1 $84$ $2$ $2$ $2$ $?$ not computed
84.48.2.s.1 $84$ $2$ $2$ $2$ $?$ not computed
84.48.3.ca.1 $84$ $2$ $2$ $3$ $?$ not computed
84.48.3.cb.1 $84$ $2$ $2$ $3$ $?$ not computed
84.48.3.ce.1 $84$ $2$ $2$ $3$ $?$ not computed
84.48.3.cf.1 $84$ $2$ $2$ $3$ $?$ not computed
$X_0(98)$ $98$ $7$ $7$ $7$ $?$ not computed
126.72.1.k.1 $126$ $3$ $3$ $1$ $?$ dimension zero
126.72.1.k.2 $126$ $3$ $3$ $1$ $?$ dimension zero
126.72.1.l.1 $126$ $3$ $3$ $1$ $?$ dimension zero
126.72.1.l.2 $126$ $3$ $3$ $1$ $?$ dimension zero
126.72.1.m.1 $126$ $3$ $3$ $1$ $?$ dimension zero
126.72.1.m.2 $126$ $3$ $3$ $1$ $?$ dimension zero
140.48.2.c.1 $140$ $2$ $2$ $2$ $?$ not computed
140.48.2.d.1 $140$ $2$ $2$ $2$ $?$ not computed
140.48.2.e.1 $140$ $2$ $2$ $2$ $?$ not computed
140.48.2.g.1 $140$ $2$ $2$ $2$ $?$ not computed
140.48.2.h.1 $140$ $2$ $2$ $2$ $?$ not computed
140.48.2.i.1 $140$ $2$ $2$ $2$ $?$ not computed
140.48.3.k.1 $140$ $2$ $2$ $3$ $?$ not computed
140.48.3.l.1 $140$ $2$ $2$ $3$ $?$ not computed
140.48.3.m.1 $140$ $2$ $2$ $3$ $?$ not computed
140.48.3.n.1 $140$ $2$ $2$ $3$ $?$ not computed
154.48.2.e.1 $154$ $2$ $2$ $2$ $?$ not computed
154.48.2.f.1 $154$ $2$ $2$ $2$ $?$ not computed
$X_0(154)$ $154$ $12$ $12$ $21$ $?$ not computed
168.48.2.p.1 $168$ $2$ $2$ $2$ $?$ not computed
168.48.2.q.1 $168$ $2$ $2$ $2$ $?$ not computed
168.48.2.r.1 $168$ $2$ $2$ $2$ $?$ not computed
168.48.2.s.1 $168$ $2$ $2$ $2$ $?$ not computed
168.48.2.bh.1 $168$ $2$ $2$ $2$ $?$ not computed
168.48.2.bi.1 $168$ $2$ $2$ $2$ $?$ not computed
168.48.2.bj.1 $168$ $2$ $2$ $2$ $?$ not computed
168.48.2.bk.1 $168$ $2$ $2$ $2$ $?$ not computed
168.48.3.gi.1 $168$ $2$ $2$ $3$ $?$ not computed
168.48.3.gj.1 $168$ $2$ $2$ $3$ $?$ not computed
168.48.3.go.1 $168$ $2$ $2$ $3$ $?$ not computed
168.48.3.gp.1 $168$ $2$ $2$ $3$ $?$ not computed
182.48.2.n.1 $182$ $2$ $2$ $2$ $?$ not computed
182.48.2.o.1 $182$ $2$ $2$ $2$ $?$ not computed
182.72.1.a.1 $182$ $3$ $3$ $1$ $?$ dimension zero
182.72.1.a.2 $182$ $3$ $3$ $1$ $?$ dimension zero
182.72.1.b.1 $182$ $3$ $3$ $1$ $?$ dimension zero
182.72.1.b.2 $182$ $3$ $3$ $1$ $?$ dimension zero
182.72.1.c.1 $182$ $3$ $3$ $1$ $?$ dimension zero
182.72.1.c.2 $182$ $3$ $3$ $1$ $?$ dimension zero
210.48.2.c.1 $210$ $2$ $2$ $2$ $?$ not computed
210.48.2.e.1 $210$ $2$ $2$ $2$ $?$ not computed
238.48.2.e.1 $238$ $2$ $2$ $2$ $?$ not computed
238.48.2.f.1 $238$ $2$ $2$ $2$ $?$ not computed
266.48.2.n.1 $266$ $2$ $2$ $2$ $?$ not computed
266.48.2.o.1 $266$ $2$ $2$ $2$ $?$ not computed
266.72.1.a.1 $266$ $3$ $3$ $1$ $?$ dimension zero
266.72.1.a.2 $266$ $3$ $3$ $1$ $?$ dimension zero
266.72.1.b.1 $266$ $3$ $3$ $1$ $?$ dimension zero
266.72.1.b.2 $266$ $3$ $3$ $1$ $?$ dimension zero
266.72.1.c.1 $266$ $3$ $3$ $1$ $?$ dimension zero
266.72.1.c.2 $266$ $3$ $3$ $1$ $?$ dimension zero
280.48.2.h.1 $280$ $2$ $2$ $2$ $?$ not computed
280.48.2.i.1 $280$ $2$ $2$ $2$ $?$ not computed
280.48.2.j.1 $280$ $2$ $2$ $2$ $?$ not computed
280.48.2.k.1 $280$ $2$ $2$ $2$ $?$ not computed
280.48.2.n.1 $280$ $2$ $2$ $2$ $?$ not computed
280.48.2.o.1 $280$ $2$ $2$ $2$ $?$ not computed
280.48.2.p.1 $280$ $2$ $2$ $2$ $?$ not computed
280.48.2.q.1 $280$ $2$ $2$ $2$ $?$ not computed
280.48.3.k.1 $280$ $2$ $2$ $3$ $?$ not computed
280.48.3.l.1 $280$ $2$ $2$ $3$ $?$ not computed
280.48.3.m.1 $280$ $2$ $2$ $3$ $?$ not computed
280.48.3.n.1 $280$ $2$ $2$ $3$ $?$ not computed
308.48.2.e.1 $308$ $2$ $2$ $2$ $?$ not computed
308.48.2.f.1 $308$ $2$ $2$ $2$ $?$ not computed
308.48.2.g.1 $308$ $2$ $2$ $2$ $?$ not computed
308.48.2.h.1 $308$ $2$ $2$ $2$ $?$ not computed
308.48.2.i.1 $308$ $2$ $2$ $2$ $?$ not computed
308.48.2.j.1 $308$ $2$ $2$ $2$ $?$ not computed
308.48.3.a.1 $308$ $2$ $2$ $3$ $?$ not computed
308.48.3.b.1 $308$ $2$ $2$ $3$ $?$ not computed
308.48.3.c.1 $308$ $2$ $2$ $3$ $?$ not computed
308.48.3.d.1 $308$ $2$ $2$ $3$ $?$ not computed
322.48.2.e.1 $322$ $2$ $2$ $2$ $?$ not computed
322.48.2.f.1 $322$ $2$ $2$ $2$ $?$ not computed