Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2+386x+1277\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z+386xz^2+1277z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+500229x+52084566\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z/{2}\Z\)
Torsion generators
\( \left(-\frac{13}{4}, \frac{9}{8}\right) \)
Integral points
None
Invariants
Conductor: | \( 42 \) | = | $2 \cdot 3 \cdot 7$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-4218578658 $ | = | $-1 \cdot 2 \cdot 3^{16} \cdot 7^{2} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{6359387729183}{4218578658} \) | = | $2^{-1} \cdot 3^{-16} \cdot 7^{-2} \cdot 97^{3} \cdot 191^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
|
Endomorphism ring: | $\Z$ | |||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
| |
Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $0.53567478369769316355008711520\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.53567478369769316355008711520\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.083141850233532\dots$ | |||
Szpiro ratio: | $7.887522502106543\dots$ |
BSD invariants
Analytic rank: | $0$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $1$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.86886186437420487794429665753\dots$ | 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);
|
Tamagawa product: | $ 4 $ = $ 1\cdot2\cdot2 $ | 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)
|
Torsion order: | $2$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
Analytic order of Ш: | $1$ ( exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
Special value: | $ L(E,1) $ ≈ $ 0.86886186437420487794429665753 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 0.868861864 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.868862 \cdot 1.000000 \cdot 4}{2^2} \approx 0.868861864$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 32 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$3$ | $2$ | $I_{16}$ | Non-split multiplicative | 1 | 1 | 16 | 16 |
$7$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
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 | 16.96.0.137 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 112 = 2^{4} \cdot 7 \), index $192$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 16 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 22 & 89 \\ 101 & 58 \end{array}\right),\left(\begin{array}{rr} 92 & 89 \\ 59 & 58 \end{array}\right),\left(\begin{array}{rr} 15 & 2 \\ 14 & 99 \end{array}\right),\left(\begin{array}{rr} 109 & 16 \\ 16 & 21 \end{array}\right),\left(\begin{array}{rr} 5 & 4 \\ 108 & 109 \end{array}\right),\left(\begin{array}{rr} 97 & 16 \\ 96 & 17 \end{array}\right)$.
The torsion field $K:=\Q(E[112])$ is a degree-$258048$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/112\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4 and 8.
Its isogeny class 42a
consists of 6 curves linked by isogenies of
degrees dividing 8.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{-2}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | 2.0.8.1-882.2-a2 |
$2$ | \(\Q(\sqrt{2}) \) | \(\Z/4\Z\) | 2.2.8.1-882.1-a3 |
$2$ | \(\Q(\sqrt{-1}) \) | \(\Z/4\Z\) | 2.0.4.1-882.1-a2 |
$4$ | 4.0.100352.4 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\zeta_{8})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | 4.2.2048.1 | \(\Z/8\Z\) | Not in database |
$4$ | \(\Q(i, \sqrt{14})\) | \(\Z/8\Z\) | Not in database |
$4$ | \(\Q(i, \sqrt{7})\) | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.40282095616.10 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.157351936.1 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.16777216.2 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.2.105226698752.2 | \(\Z/16\Z\) | Not in database |
$8$ | 8.0.30840979456.2 | \(\Z/16\Z\) | Not in database |
$8$ | 8.2.6805279152.1 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | 16.0.1622647227216566419456.13 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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 | 3 | 7 |
---|---|---|---|
Reduction type | split | nonsplit | nonsplit |
$\lambda$-invariant(s) | 1 | 0 | 0 |
$\mu$-invariant(s) | 2 | 0 | 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 $0$.