Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-3023x+23447\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-3023xz^2+23447z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-48363x+1452262\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(-5, 198\right)\) | \(\left(93, 688\right)\) |
$\hat{h}(P)$ | ≈ | $0.69562050516327303901248105489$ | $0.79464082583441028543203327836$ |
Torsion generators
\( \left(51, -26\right) \)
Integral points
\( \left(-57, 118\right) \), \( \left(-57, -62\right) \), \( \left(-47, 268\right) \), \( \left(-47, -222\right) \), \( \left(-33, 310\right) \), \( \left(-33, -278\right) \), \( \left(-5, 198\right) \), \( \left(-5, -194\right) \), \( \left(3, 118\right) \), \( \left(3, -122\right) \), \( \left(51, -26\right) \), \( \left(55, 118\right) \), \( \left(55, -174\right) \), \( \left(67, 310\right) \), \( \left(67, -378\right) \), \( \left(93, 688\right) \), \( \left(93, -782\right) \), \( \left(151, 1654\right) \), \( \left(151, -1806\right) \), \( \left(163, 1878\right) \), \( \left(163, -2042\right) \), \( \left(345, 6148\right) \), \( \left(345, -6494\right) \), \( \left(1227, 42310\right) \), \( \left(1227, -43538\right) \), \( \left(1563, 60958\right) \), \( \left(1563, -62522\right) \), \( \left(94243, 28884438\right) \), \( \left(94243, -28978682\right) \), \( \left(52590813, 381359772208\right) \), \( \left(52590813, -381412363022\right) \)
Invariants
Conductor: | \( 83790 \) | = | $2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $1545060787200 $ | = | $2^{10} \cdot 3^{3} \cdot 5^{2} \cdot 7^{6} \cdot 19 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( \frac{961504803}{486400} \) | = | $2^{-10} \cdot 3^{3} \cdot 5^{-2} \cdot 7^{3} \cdot 19^{-1} \cdot 47^{3}$ | 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: | $1.0313879082704576725344240291\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $-0.21622023842422640286706365185\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $0.9298072147700007\dots$ | |||
Szpiro ratio: | $3.1452973727865605\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $0.49458403036259441389823554965\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.74857334052363972537105435204\dots$ | 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: | $ 160 $ = $ ( 2 \cdot 5 )\cdot2\cdot2\cdot2^{2}\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)
|
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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
Special value: | $ L^{(2)}(E,1)/2! $ ≈ $ 14.809296791126902292364910064 $ | 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
$\displaystyle 14.809296791 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.748573 \cdot 0.494584 \cdot 160}{2^2} \approx 14.809296791$
Modular invariants
Modular form 83790.2.a.cr
For more coefficients, see the Downloads section to the right.
Modular degree: | 184320 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $10$ | $I_{10}$ | Split multiplicative | -1 | 1 | 10 | 10 |
$3$ | $2$ | $III$ | Additive | 1 | 2 | 3 | 0 |
$5$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$7$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$19$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
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 | 2.3.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 456 = 2^{3} \cdot 3 \cdot 19 \), index $12$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 308 & 1 \\ 151 & 0 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 194 & 1 \\ 359 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 2 & 5 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 453 & 4 \\ 452 & 5 \end{array}\right),\left(\begin{array}{rr} 229 & 4 \\ 2 & 9 \end{array}\right),\left(\begin{array}{rr} 289 & 172 \\ 56 & 399 \end{array}\right)$.
The torsion field $K:=\Q(E[456])$ is a degree-$756449280$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/456\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 83790cz
consists of 2 curves linked by isogenies of
degree 2.
Twists
The minimal quadratic twist of this elliptic curve is 1710a1, its twist by $21$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{57}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$4$ | 4.4.1608768.4 | \(\Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.8.934316546494464.12 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | 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 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | add | nonsplit | add | ord | ss | ord | split | ord | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | 10 | - | 2 | - | 2 | 2,2 | 2 | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2,2 |
$\mu$-invariant(s) | 0 | - | 0 | - | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.