This curve corresponds to a sporadic degree 3 point on $X_1(21)$. It is the unique elliptic curve over $\Q$ that has a point of order 21 defined over a cubic field [10.4310/MRL.2016.v23.n1.a12, MR:3512885, arXiv:1211.2188].
Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-5x+5\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-5xz^2+5z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-75x+262\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{3}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(1, 0)$ | $0$ | $3$ |
Integral points
\( \left(1, 0\right) \), \( \left(1, -2\right) \)
Invariants
| Conductor: | $N$ | = | \( 162 \) | = | $2 \cdot 3^{4}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $\Delta$ | = | $-648$ | = | $-1 \cdot 2^{3} \cdot 3^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | $j$ | = | \( -\frac{140625}{8} \) | = | $-1 \cdot 2^{-3} \cdot 3^{2} \cdot 5^{6}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $-0.70358699565001647072183612100$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-1.0697910918727197011869178666$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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| $abc$ quality: | $Q$ | ≈ | $1.1780968639246816$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.2121393486428227$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Mordell-Weil rank: | $r$ | = | $ 0$ | comment: Rank
sage: E.rank()
gp: [lower,upper] = ellrank(E)
magma: Rank(E);
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $\Omega$ | ≈ | $5.0519074123511111955585639056$ | comment: Real Period
sage: E.period_lattice().omega()
gp: if(E.disc>0,2,1)*E.omega[1]
magma: (Discriminant(E) gt 0 select 2 else 1) * RealPeriod(E);
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 3 $ = $ 3\cdot1 $ | comment: Tamagawa numbers
sage: E.tamagawa_numbers()
gp: gr=ellglobalred(E); [[gr[4][i,1],gr[5][i][4]] | i<-[1..#gr[4][,1]]]
magma: TamagawaNumbers(E);
oscar: tamagawa_numbers(E)
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $3$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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| Special value: | $ L(E,1)$ | ≈ | $1.6839691374503703985195213019 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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BSD formula
$\displaystyle 1.683969137 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 5.051907 \cdot 1.000000 \cdot 3}{3^2} \approx 1.683969137$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 6 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 2 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $3$ | $1$ | $II$ | additive | 1 | 4 | 4 | 0 |
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 |
|---|---|---|
| $2$ | 2G | 8.2.0.1 |
| $3$ | 3B.1.1 | 3.8.0.1 |
| $7$ | 7B.2.1 | 7.16.0.1 |
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 $768$, genus $21$, and generators
$\left(\begin{array}{rr} 281 & 392 \\ 112 & 393 \end{array}\right),\left(\begin{array}{rr} 1 & 426 \\ 42 & 253 \end{array}\right),\left(\begin{array}{rr} 1 & 216 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 462 & 1 \end{array}\right),\left(\begin{array}{rr} 85 & 42 \\ 189 & 43 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 168 & 1 \end{array}\right),\left(\begin{array}{rr} 337 & 168 \\ 336 & 169 \end{array}\right),\left(\begin{array}{rr} 295 & 42 \\ 231 & 211 \end{array}\right),\left(\begin{array}{rr} 22 & 321 \\ 189 & 169 \end{array}\right),\left(\begin{array}{rr} 1 & 42 \\ 0 & 73 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 420 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 168 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 463 & 282 \\ 420 & 295 \end{array}\right),\left(\begin{array}{rr} 177 & 10 \\ 448 & 417 \end{array}\right)$.
The torsion field $K:=\Q(E[504])$ is a degree-$15676416$ 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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 81 = 3^{4} \) |
| $3$ | additive | $8$ | \( 1 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 7 and 21.
Its isogeny class 162b
consists of 4 curves linked by isogenies of
degrees dividing 21.
Twists
The minimal quadratic twist of this elliptic curve is 162c2, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.648.1 | \(\Z/6\Z\) | not in database |
| $3$ | \(\Q(\zeta_{9})^+\) | \(\Z/21\Z\) | 3.3.81.1-8.1-a1 |
| $6$ | 6.0.3359232.4 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.177147.2 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $9$ | 9.3.74384733888.1 | \(\Z/9\Z\) | not in database |
| $9$ | 9.3.272097792.1 | \(\Z/42\Z\) | not in database |
| $12$ | 12.2.5777633090469888.10 | \(\Z/12\Z\) | not in database |
| $18$ | 18.0.1062353018033006514536448.1 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.37907050706572935168.1 | \(\Z/2\Z \oplus \Z/42\Z\) | not in database |
| $18$ | 18.0.4052555153018976267.1 | \(\Z/3\Z \oplus \Z/21\Z\) | not in database |
We only show fields where the torsion growth is primitive.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 |
|---|---|---|---|---|
| Reduction type | split | add | ss | ord |
| $\lambda$-invariant(s) | 4 | - | 0,0 | 0 |
| $\mu$-invariant(s) | 0 | - | 0,0 | 0 |
All Iwasawa $\lambda$ and $\mu$-invariants for primes $p\ge 11$ of good reduction are zero.
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
All $p$-adic regulators are identically $1$ since the rank is $0$.