This is a model for the modular curve $X_1(14)$.
Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-xz^2\)
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(dehomogenize, simplify) |
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\(y^2=x^3-675x+13662\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{6}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1, 0)$ | $0$ | $6$ |
Integral points
\( \left(-1, 0\right) \), \( \left(0, 0\right) \), \( \left(0, -1\right) \), \( \left(1, 0\right) \), \( \left(1, -2\right) \)
Invariants
| Conductor: | $N$ | = | \( 14 \) | = | $2 \cdot 7$ |
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| Discriminant: | $\Delta$ | = | $-28$ | = | $-1 \cdot 2^{2} \cdot 7 $ |
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| j-invariant: | $j$ | = | \( -\frac{15625}{28} \) | = | $-1 \cdot 2^{-2} \cdot 5^{6} \cdot 7^{-1}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $-1.0320848985249978885230789428$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0320848985249978885230789428$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0171207213859017$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.193459101196728$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $5.9440258682006497025087150302$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 2 $ = $ 2\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $6$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.33022365934448053902826194612 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.330223659 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 5.944026 \cdot 1.000000 \cdot 2}{6^2} \\ & \approx 0.330223659\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 3 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 3 |
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Local data at primes of bad reduction
This elliptic curve is semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $7$ | $1$ | $I_{1}$ | split multiplicative | -1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$ except those listed in the table below.
| prime $\ell$ | mod-$\ell$ image | $\ell$-adic image |
|---|---|---|
| $2$ | 2B | 8.6.0.1 |
| $3$ | 3B.1.1 | 9.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \), index $864$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 253 & 36 \\ 10 & 361 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 36 & 1 \end{array}\right),\left(\begin{array}{rr} 281 & 36 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 127 & 36 \\ 262 & 361 \end{array}\right),\left(\begin{array}{rr} 469 & 36 \\ 468 & 37 \end{array}\right),\left(\begin{array}{rr} 19 & 36 \\ 216 & 91 \end{array}\right),\left(\begin{array}{rr} 160 & 9 \\ 47 & 22 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 14 & 253 \end{array}\right)$.
The torsion field $K:=\Q(E[504])$ is a degree-$13934592$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/504\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 7 \) |
| $7$ | split multiplicative | $8$ | \( 2 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 6, 9 and 18.
Its isogeny class 14.a
consists of 6 curves linked by isogenies of
degrees dividing 18.
Twists
This elliptic curve is its own minimal quadratic twist.
Growth of torsion in number fields
The number fields $K$ of degree less than 24 such that $E(K)_{\rm tors}$ is strictly larger than $E(\Q)_{\rm tors}$ $\cong \Z/{6}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \oplus \Z/6\Z\) | 2.0.7.1-28.2-a6 |
| $3$ | \(\Q(\zeta_{7})^+\) | \(\Z/18\Z\) | 3.3.49.1-56.1-a5 |
| $4$ | 4.2.448.1 | \(\Z/12\Z\) | not in database |
| $6$ | 6.0.1037232.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.21168.1 | \(\Z/18\Z\) | not in database |
| $6$ | \(\Q(\zeta_{7})\) | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $8$ | 8.0.9834496.2 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $8$ | 8.0.120472576.1 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $12$ | 12.0.52716660869376.1 | \(\Z/6\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.1075850221824.1 | \(\Z/2\Z \oplus \Z/18\Z\) | not in database |
| $12$ | 12.6.10578455953408.1 | \(\Z/36\Z\) | not in database |
| $16$ | deg 16 | \(\Z/24\Z\) | not in database |
| $18$ | 18.0.1115906277282951168.1 | \(\Z/3\Z \oplus \Z/18\Z\) | not in database |
We only show fields where the torsion growth is primitive. For fields not in the database, click on the degree shown to reveal the defining polynomial.
Iwasawa invariants
| $p$ | 2 | 3 | 7 |
|---|---|---|---|
| Reduction type | nonsplit | ord | split |
| $\lambda$-invariant(s) | 0 | 0 | 1 |
| $\mu$-invariant(s) | 0 | 0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 5$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Additional information
This curve $E$ also parametrizes pairs $(R,T)$ where $R$ is a rational rectangle, $T$ is a Pythagorean triangle, and $R,T$ have the same perimeter and the same area. (That is, $E$ is birational with the curve of $(a:b:c:x:y) \in {\bf P}^4$ with $a^2+b^2=c^2$, $a+b+c=2x+2y$, and $ab/2=xy$.) Unfortunately the six rational points on $E$ all yield degenerate solutions. [Noted in passing in Richard Guy's paper "My Favorite Elliptic Curve: A Tale of Two Types of Triangles", Amer. Math. Monthly 102 #9 (Nov. 1995), 771-781.]