This is a model for the modular curve $X_1(14)$.
Minimal Weierstrass equation
\(y^2+xy+y=x^3-x\) 
Mordell-Weil group structure
\(\Z/{6}\Z\)
Torsion generators
\( \left(1, 0\right) \)
Integral points
\( \left(-1, 0\right) \), \( \left(0, 0\right) \), \( \left(0, -1\right) \), \( \left(1, 0\right) \), \( \left(1, -2\right) \)
Invariants
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sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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| Conductor: | \( 14 \) | = | \(2 \cdot 7\) |
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sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
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| Discriminant: | \(-28 \) | = | \(-1 \cdot 2^{2} \cdot 7 \) |
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sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
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| j-invariant: | \( -\frac{15625}{28} \) | = | \(-1 \cdot 2^{-2} \cdot 5^{6} \cdot 7^{-1}\) |
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
| Faltings height: | \(-1.0320848985249978885230789428\dots\) | ||
| Stable Faltings height: | \(-1.0320848985249978885230789428\dots\) | ||
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: | \(5.9440258682006497025087150302\dots\) | ||
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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: | \( 2 \) = \( 2\cdot1 \) | ||
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sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
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| Torsion order: | \(6\) | ||
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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: | 3 | ||
| \( \Gamma_0(N) \)-optimal: | no | ||
| Manin constant: | 3 | ||
Special L-value
\( L(E,1) \) ≈ \( 0.33022365934448053902826194612222809986 \)
Local data
This elliptic curve is semistable. There are 2 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(N\)) | ord(\(\Delta\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|
| \(2\) | \(2\) | \(I_{2}\) | Non-split multiplicative | 1 | 1 | 2 | 2 |
| \(7\) | \(1\) | \(I_{1}\) | Split multiplicative | -1 | 1 | 1 | 1 |
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 X16.
This subgroup is the pull-back of the subgroup of $\GL(2,\Z_2/2^3\Z_2)$ generated by $\left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 0 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5 & 0 \\ 2 & 1 \end{array}\right)$ and has index 6.
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 |
| \(3\) | B.1.1 |
$p$-adic data
$p$-adic regulators
All \(p\)-adic regulators are identically \(1\) since the rank is \(0\).
Iwasawa invariants
| $p$ | 2 | 3 | 7 |
|---|---|---|---|
| Reduction type | nonsplit | ordinary | 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.
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.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{-7}) \) | \(\Z/2\Z \times \Z/6\Z\) | 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 | \(\Z/3\Z \times \Z/6\Z\) | Not in database |
| $6$ | 6.0.21168.1 | \(\Z/18\Z\) | Not in database |
| $6$ | \(\Q(\zeta_{7})\) | \(\Z/2\Z \times \Z/18\Z\) | Not in database |
| $8$ | 8.0.9834496.2 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $8$ | 8.0.120472576.1 | \(\Z/2\Z \times \Z/12\Z\) | Not in database |
| $12$ | 12.0.52716660869376.1 | \(\Z/6\Z \times \Z/6\Z\) | Not in database |
| $12$ | 12.0.1075850221824.1 | \(\Z/2\Z \times \Z/18\Z\) | 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 | \(\Z/3\Z \times \Z/18\Z\) | 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.
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.]