This is a model for the modular curve $X_0(17)$.
Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-x-14\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-xz^2-14z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-11x-890\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{4}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(7, 13)$ | $0$ | $4$ |
Integral points
\( \left(7, 13\right) \), \( \left(7, -21\right) \)
Invariants
| Conductor: | $N$ | = | \( 17 \) | = | $17$ |
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| Discriminant: | $\Delta$ | = | $-83521$ | = | $-1 \cdot 17^{4} $ |
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| j-invariant: | $j$ | = | \( -\frac{35937}{83521} \) | = | $-1 \cdot 3^{3} \cdot 11^{3} \cdot 17^{-4}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.37663612094134348095743300423$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.37663612094134348095743300423$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1807067659885138$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.631276933576505$ | |||
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: | $\Omega$ | ≈ | $1.5470797535511201732095790050$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 4 $ = $ 2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L(E,1)$ | ≈ | $0.38676993838778004330239475124 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 0.386769938 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 1.547080 \cdot 1.000000 \cdot 4}{4^2} \\ & \approx 0.386769938\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 1 |
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is semistable. There is only one prime $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $17$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
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 | 16.96.0.32 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1088 = 2^{6} \cdot 17 \), index $1536$, genus $53$, and generators
$\left(\begin{array}{rr} 191 & 478 \\ 974 & 279 \end{array}\right),\left(\begin{array}{rr} 1 & 64 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 174 & 863 \\ 739 & 498 \end{array}\right),\left(\begin{array}{rr} 9 & 124 \\ 756 & 745 \end{array}\right),\left(\begin{array}{rr} 513 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1025 & 64 \\ 1024 & 65 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 64 & 1 \end{array}\right),\left(\begin{array}{rr} 57 & 16 \\ 176 & 393 \end{array}\right)$.
The torsion field $K:=\Q(E[1088])$ is a degree-$320864256$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1088\Z)$.
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 |
|---|---|---|---|
| $2$ | good | $2$ | \( 1 \) |
| $17$ | split multiplicative | $18$ | \( 1 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 17a
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{4}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/4\Z\) | 2.0.4.1-289.2-a1 |
| $4$ | \(\Q(\zeta_{8})\) | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $4$ | 4.2.1156.1 | \(\Z/8\Z\) | not in database |
| $8$ | 8.0.5473632256.2 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.0.21381376.2 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $8$ | 8.2.182660427.1 | \(\Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | 16.0.29960650073923649536.2 | \(\Z/4\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/16\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | 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 | 17 |
|---|---|---|
| Reduction type | ord | split |
| $\lambda$-invariant(s) | 0 | 1 |
| $\mu$-invariant(s) | 2 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 3$ of good reduction are zero.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.
Additional information
This is the only elliptic curve over $\Q$ of prime conductor $N$ and discriminant $\Delta = \pm N^4$, and one of only four prime-conductor curves of discriminant $\pm N^e$ with $e > 2$: the others are the modular curves $X_0(N)$ for N=11 [11.a2] and N=19 [19.a2], and the curve [37.b2] (with $e=5,3,3$ respectively -- in each case, as here with $(N,e)=(17,4)$, the exponent $e$ is the numerator of $(N-1)/12$).