This is a model for the modular curve $X_0(17)$.
Minimal Weierstrass equation
\(y^2+xy+y=x^3-x^2-x-14\)
Mordell-Weil group structure
\(\Z/{4}\Z\)
Torsion generators
\( \left(7, 13\right) \)
Integral points
\( \left(7, 13\right) \), \( \left(7, -21\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 17 \) | = | \(17\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-83521 \) | = | \(-1 \cdot 17^{4} \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{35937}{83521} \) | = | \(-1 \cdot 3^{3} \cdot 11^{3} \cdot 17^{-4}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
BSD invariants
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sage: E.rank()
magma: Rank(E);
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| Analytic rank: | \(0\) | ||
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sage: E.regulator()
magma: Regulator(E);
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| Regulator: | \(1\) | ||
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sage: E.period_lattice().omega()
gp: E.omega[1]
magma: RealPeriod(E);
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| Real period: | \(1.5470797535511201732095790050\) | ||
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sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
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| Tamagawa product: | \( 4 \) = \( 2^{2} \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(4\) | ||
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sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Analytic order of Ш: | \(1\) (exact) | ||
Modular invariants
For more coefficients, see the Downloads section to the right.
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sage: E.modular_degree()
magma: ModularDegree(E);
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| Modular degree: | 1 | ||
| \( \Gamma_0(N) \)-optimal: | yes | ||
| Manin constant: | 1 | ||
Special L-value
\( L(E,1) \) ≈ \( 0.38676993838778004330239475124309554740 \)
Local data
This elliptic curve is semistable. There is only one prime of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(17\) | \(4\) | \(I_{4}\) | Split multiplicative | -1 | 1 | 4 | 4 |
Galois representations
The image of the 2-adic representation attached to this elliptic curve is the subgroup of $\GL(2,\Z_2)$ with Rouse label X221a.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^4\Z_2)$ generated by $\left(\begin{array}{rr} 9 & 9 \\ 12 & 3 \end{array}\right),\left(\begin{array}{rr} 5 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 13 & 10 \\ 4 & 1 \end{array}\right)$ and has index 96.
The mod \( p \) Galois representation has maximal image \(\GL(2,\F_p)\) for all primes \( p \) except those listed.
| prime | Image of Galois representation |
|---|---|
| \(2\) | B |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
| $p$ | 2 | 17 |
|---|---|---|
| Reduction type | ordinary | 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.
Isogenies
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class 17.a
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \times \Z/4\Z\) | 2.0.4.1-289.2-a1 |
| $4$ | \(\Q(\zeta_{8})\) | \(\Z/4\Z \times \Z/4\Z\) | Not in database |
| $4$ | 4.2.1156.1 | \(\Z/8\Z\) | Not in database |
| $8$ | 8.0.5473632256.2 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $8$ | 8.0.21381376.2 | \(\Z/2\Z \times \Z/8\Z\) | Not in database |
| $8$ | 8.2.182660427.1 | \(\Z/12\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/4\Z \times \Z/8\Z\) | Not in database |
| $16$ | 16.0.29960650073923649536.2 | \(\Z/4\Z \times \Z/8\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/16\Z\) | Not in database |
| $16$ | Deg 16 | \(\Z/2\Z \times \Z/12\Z\) | 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.
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 $(N-1)/12$).