Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-11842820x+15197648400\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-11842820xz^2+15197648400z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-15348294747x+709107528634614\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{10}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(1672, 7612\right)\) |
$\hat{h}(P)$ | ≈ | $1.8900143657474552481085852362$ |
Torsion generators
\( \left(1540, 23980\right) \)
Integral points
\( \left(-3960, 1980\right) \), \( \left(-3410, 127930\right) \), \( \left(-3410, -124520\right) \), \( \left(-1760, 175780\right) \), \( \left(-1760, -174020\right) \), \( \left(-440, 142780\right) \), \( \left(-440, -142340\right) \), \( \left(1540, 23980\right) \), \( \left(1540, -25520\right) \), \( \left(1672, 7612\right) \), \( \left(1672, -9284\right) \), \( \left(2290, 8230\right) \), \( \left(2290, -10520\right) \), \( \left(2440, 27580\right) \), \( \left(2440, -30020\right) \), \( \left(2800, 61780\right) \), \( \left(2800, -64580\right) \), \( \left(3880, 164380\right) \), \( \left(3880, -168260\right) \), \( \left(6040, 401980\right) \), \( \left(6040, -408020\right) \), \( \left(12040, 1265980\right) \), \( \left(12040, -1278020\right) \), \( \left(38476, 7498732\right) \), \( \left(38476, -7537208\right) \), \( \left(60040, 14657980\right) \), \( \left(60040, -14718020\right) \)
Invariants
Conductor: | \( 29370 \) | = | $2 \cdot 3 \cdot 5 \cdot 11 \cdot 89$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
|
Discriminant: | $6511602493440000000000 $ | = | $2^{20} \cdot 3^{10} \cdot 5^{10} \cdot 11^{2} \cdot 89 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{183691516586815867994210881}{6511602493440000000000} \) | = | $2^{-20} \cdot 3^{-10} \cdot 5^{-10} \cdot 11^{-2} \cdot 29^{3} \cdot 89^{-1} \cdot 19601909^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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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: | $2.9562987849383155322898892919\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $2.9562987849383155322898892919\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9970832623240626\dots$ | |||
Szpiro ratio: | $5.878391615875246\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $1.8900143657474552481085852362\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.13266933675265705122314127868\dots$ | 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: | $ 4000 $ = $ ( 2^{2} \cdot 5 )\cdot( 2 \cdot 5 )\cdot( 2 \cdot 5 )\cdot2\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: | $10$ | 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);
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Special value: | $ L'(E,1) $ ≈ $ 10.029878094268346829965462371 $ | 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
$\displaystyle 10.029878094 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.132669 \cdot 1.890014 \cdot 4000}{10^2} \approx 10.029878094$
Modular invariants
Modular form 29370.2.a.bm
For more coefficients, see the Downloads section to the right.
Modular degree: | 2560000 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
|
$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $20$ | $I_{20}$ | Split multiplicative | -1 | 1 | 20 | 20 |
$3$ | $10$ | $I_{10}$ | Split multiplicative | -1 | 1 | 10 | 10 |
$5$ | $10$ | $I_{10}$ | Split multiplicative | -1 | 1 | 10 | 10 |
$11$ | $2$ | $I_{2}$ | Split multiplicative | -1 | 1 | 2 | 2 |
$89$ | $1$ | $I_{1}$ | Non-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 |
$5$ | 5B.1.1 | 5.24.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 19580 = 2^{2} \cdot 5 \cdot 11 \cdot 89 \), index $288$, genus $5$, and generators
$\left(\begin{array}{rr} 11 & 16 \\ 19340 & 19231 \end{array}\right),\left(\begin{array}{rr} 11751 & 20 \\ 190 & 1267 \end{array}\right),\left(\begin{array}{rr} 1 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 19561 & 20 \\ 19560 & 21 \end{array}\right),\left(\begin{array}{rr} 9791 & 20 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 10 & 101 \end{array}\right),\left(\begin{array}{rr} 1116 & 5 \\ 7875 & 19566 \end{array}\right),\left(\begin{array}{rr} 8901 & 20 \\ 10690 & 201 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 20 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[19580])$ is a degree-$131006177280000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/19580\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 5 and 10.
Its isogeny class 29370.bm
consists of 4 curves linked by isogenies of
degrees dividing 10.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{10}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{89}) \) | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$4$ | 4.0.4307600.1 | \(\Z/20\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/20\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/20\Z\) | Not in database |
$8$ | deg 8 | \(\Z/30\Z\) | Not in database |
$16$ | deg 16 | \(\Z/40\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/30\Z\) | Not in database |
$20$ | 20.0.21730670860452323943463871650446339901635391611358642578125.3 | \(\Z/5\Z \oplus \Z/10\Z\) | 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 | 89 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | split | split | ord | split | ord | ord | ss | ord | ss | ord | ord | ord | ord | ord | nonsplit |
$\lambda$-invariant(s) | 3 | 4 | 2 | 1 | 4 | 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 | 0 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.