Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-3877875x+2939267250\) | (homogenize, simplify) |
\(y^2z=x^3-3877875xz^2+2939267250z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-3877875x+2939267250\) | (homogenize, minimize) |
Mordell-Weil group structure
trivial
Integral points
None
Invariants
Conductor: | \( 32400 \) | = | $2^{4} \cdot 3^{4} \cdot 5^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-4353564672000000 $ | = | $-1 \cdot 2^{19} \cdot 3^{12} \cdot 5^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( -\frac{189613868625}{128} \) | = | $-1 \cdot 2^{-7} \cdot 3^{3} \cdot 5^{3} \cdot 383^{3}$ | 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: | $2.3165403599886905242460746573\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $-0.27993806545641466386678236769\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
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: | $0.36156594183341731117091530343\dots$ | 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: | $ 8 $ = $ 2^{2}\cdot1\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: | $1$ | 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) $ ≈ $ 2.8925275346673384893673224274 $ | 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
$\displaystyle 2.892527535 \approx L(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.361566 \cdot 1.000000 \cdot 8}{1^2} \approx 2.892527535$
Modular invariants
Modular form 32400.2.a.cv
For more coefficients, see the Downloads section to the right.
Modular degree: | 435456 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{11}^{*}$ | Additive | -1 | 4 | 19 | 7 |
$3$ | $1$ | $II^{*}$ | Additive | 1 | 4 | 12 | 0 |
$5$ | $2$ | $I_0^{*}$ | Additive | 1 | 2 | 6 | 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 | 3.4.0.1 |
$7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \), index $768$, genus $21$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 1470 & 1 \end{array}\right),\left(\begin{array}{rr} 281 & 1400 \\ 1120 & 1401 \end{array}\right),\left(\begin{array}{rr} 1 & 930 \\ 1050 & 1261 \end{array}\right),\left(\begin{array}{rr} 503 & 0 \\ 0 & 2519 \end{array}\right),\left(\begin{array}{rr} 209 & 1470 \\ 1785 & 2309 \end{array}\right),\left(\begin{array}{rr} 1 & 720 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 1680 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1471 & 1290 \\ 420 & 2311 \end{array}\right),\left(\begin{array}{rr} 2101 & 1050 \\ 2205 & 1051 \end{array}\right),\left(\begin{array}{rr} 526 & 825 \\ 2205 & 1681 \end{array}\right),\left(\begin{array}{rr} 505 & 2016 \\ 504 & 505 \end{array}\right),\left(\begin{array}{rr} 1081 & 1410 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1680 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2016 & 1 \end{array}\right),\left(\begin{array}{rr} 841 & 1680 \\ 840 & 1681 \end{array}\right),\left(\begin{array}{rr} 39 & 1070 \\ 560 & 2319 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 420 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[2520])$ is a degree-$7524679680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2520\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 7 and 21.
Its isogeny class 32400bq
consists of 4 curves linked by isogenies of
degrees dividing 21.
Twists
The minimal quadratic twist of this elliptic curve is 162c3, its twist by $60$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{15}) \) | \(\Z/3\Z\) | Not in database |
$3$ | 3.1.648.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.3359232.4 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.0.23328000.1 | \(\Z/3\Z\) | Not in database |
$6$ | 6.6.882165816000.1 | \(\Z/7\Z\) | Not in database |
$6$ | 6.2.2519424000.3 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | 12.0.4897760256000000.3 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$12$ | 12.12.7003948742270512704000000.4 | \(\Z/21\Z\) | Not in database |
$18$ | 18.6.2689081076896047739920384000000000.2 | \(\Z/9\Z\) | Not in database |
$18$ | 18.0.6908559991272917434368000000000.2 | \(\Z/6\Z\) | Not in database |
$18$ | 18.6.2811969607655988580016367599616000000000.1 | \(\Z/14\Z\) | 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 | add | add | add | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | - | - | - | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
$\mu$-invariant(s) | - | - | - | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.