Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3+x^2-652548353x-6416258962177\) | (homogenize, simplify) |
\(y^2z=x^3+x^2z-652548353xz^2-6416258962177z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-52856416620x-4677294214177200\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{72599622167}{1413721}, \frac{16426994553813088}{1680914269}\right)\) |
$\hat{h}(P)$ | ≈ | $20.586706695436404296011372267$ |
Integral points
None
Invariants
Conductor: | \( 78400 \) | = | $2^{6} \cdot 5^{2} \cdot 7^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $-188900999168000 $ | = | $-1 \cdot 2^{18} \cdot 5^{3} \cdot 7^{8} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( -162677523113838677 \) | = | $-1 \cdot 7 \cdot 137^{3} \cdot 2083^{3}$ | 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: | $3.2831605738749430796791588227\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $0.54380689222295781849955231158\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
||
$abc$ quality: | $1.0734398000116747\dots$ | |||
Szpiro ratio: | $6.43349579453399\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $20.586706695436404296011372267\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.014935383703168251758324187972\dots$ | 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: | $ 24 $ = $ 2^{2}\cdot2\cdot3 $ | 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: | $1$ ( rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
|
Special value: | $ L'(E,1) $ ≈ $ 7.3792887283422145630842946934 $ | 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
$\displaystyle 7.379288728 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.014935 \cdot 20.586707 \cdot 24}{1^2} \approx 7.379288728$
Modular invariants
Modular form 78400.2.a.hz
For more coefficients, see the Downloads section to the right.
Modular degree: | 6365184 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{8}^{*}$ | Additive | -1 | 6 | 18 | 0 |
$5$ | $2$ | $III$ | Additive | -1 | 2 | 3 | 0 |
$7$ | $3$ | $IV^{*}$ | Additive | 1 | 2 | 8 | 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 |
---|---|---|
$37$ | 37B.8.2 | 37.114.4.2 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10360 = 2^{3} \cdot 5 \cdot 7 \cdot 37 \), index $2736$, genus $97$, and generators
$\left(\begin{array}{rr} 1 & 3256 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 148 & 1 \end{array}\right),\left(\begin{array}{rr} 10255 & 7326 \\ 10323 & 9577 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 5180 & 1 \end{array}\right),\left(\begin{array}{rr} 5181 & 5180 \\ 5180 & 5181 \end{array}\right),\left(\begin{array}{rr} 9323 & 0 \\ 0 & 8287 \end{array}\right),\left(\begin{array}{rr} 1 & 28 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 5180 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1925 & 3256 \\ 1924 & 1925 \end{array}\right),\left(\begin{array}{rr} 113 & 4144 \\ 148 & 1 \end{array}\right),\left(\begin{array}{rr} 7769 & 7770 \\ 7770 & 7769 \end{array}\right),\left(\begin{array}{rr} 85 & 3108 \\ 148 & 3209 \end{array}\right),\left(\begin{array}{rr} 5919 & 0 \\ 0 & 6659 \end{array}\right),\left(\begin{array}{rr} 1 & 2590 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 4848 & 2923 \\ 4477 & 1259 \end{array}\right),\left(\begin{array}{rr} 5179 & 0 \\ 0 & 10359 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2590 & 1 \end{array}\right),\left(\begin{array}{rr} 10285 & 10286 \\ 8769 & 8065 \end{array}\right)$.
The torsion field $K:=\Q(E[10360])$ is a degree-$989913415680$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10360\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
37.
Its isogeny class 78400jm
consists of 2 curves linked by isogenies of
degree 37.
Twists
The minimal quadratic twist of this elliptic curve is 1225h2, its twist by $56$.
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 |
---|---|---|---|
$3$ | 3.1.980.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.19208000.2 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | deg 8 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | 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 | ord | add | add | ss | ord | ord | ord | ord | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | - | 3 | - | - | 1,1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 |
$\mu$-invariant(s) | - | 0 | - | - | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0,0 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.