Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2-757071616x-7990025437184\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z-757071616xz^2-7990025437184z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-61322800923x-5824912512109878\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{31424585127487380306}{988175634138121}, \frac{9468026884923925985105346250}{31063558972972343684581}\right)\) |
$\hat{h}(P)$ | ≈ | $41.386998318937255229093559622$ |
Torsion generators
\( \left(-16639, 0\right) \)
Integral points
\( \left(-16639, 0\right) \)
Invariants
Conductor: | \( 38640 \) | = | $2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 23$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $190117365759317606400000000 $ | = | $2^{21} \cdot 3^{6} \cdot 5^{8} \cdot 7^{12} \cdot 23 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{11715873038622856702991202049}{46415372499833400000000} \) | = | $2^{-9} \cdot 3^{-6} \cdot 5^{-8} \cdot 7^{-12} \cdot 23^{-1} \cdot 107^{3} \cdot 239^{3} \cdot 88813^{3}$ | 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: | $3.8995730225160066525925668262\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $3.2064258419560613431753347047\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0367772299224567\dots$ | |||
Szpiro ratio: | $6.906666284023863\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $41.386998318937255229093559622\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.028788484791623756627188673657\dots$ | 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: | $ 16 $ = $ 2\cdot2\cdot2\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)
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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);
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Special value: | $ L'(E,1) $ ≈ $ 4.7658758867027326157077150831 $ | 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
$\displaystyle 4.765875887 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.028788 \cdot 41.386998 \cdot 16}{2^2} \approx 4.765875887$
Modular invariants
Modular form 38640.2.a.k
For more coefficients, see the Downloads section to the right.
Modular degree: | 15925248 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $2$ | $I_{13}^{*}$ | Additive | -1 | 4 | 21 | 9 |
$3$ | $2$ | $I_{6}$ | Non-split multiplicative | 1 | 1 | 6 | 6 |
$5$ | $2$ | $I_{8}$ | Non-split multiplicative | 1 | 1 | 8 | 8 |
$7$ | $2$ | $I_{12}$ | Non-split multiplicative | 1 | 1 | 12 | 12 |
$23$ | $1$ | $I_{1}$ | 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 | 8.12.0.7 |
$3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \), index $384$, genus $5$, and generators
$\left(\begin{array}{rr} 2761 & 24 \\ 2220 & 289 \end{array}\right),\left(\begin{array}{rr} 2094 & 2761 \\ 3059 & 3704 \end{array}\right),\left(\begin{array}{rr} 15 & 106 \\ 2558 & 2363 \end{array}\right),\left(\begin{array}{rr} 3208 & 21 \\ 1059 & 3490 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 12 & 145 \end{array}\right),\left(\begin{array}{rr} 1933 & 24 \\ 2576 & 3221 \end{array}\right),\left(\begin{array}{rr} 962 & 3861 \\ 921 & 3830 \end{array}\right),\left(\begin{array}{rr} 3841 & 24 \\ 3840 & 25 \end{array}\right),\left(\begin{array}{rr} 1 & 24 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 24 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[3864])$ is a degree-$103413252096$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/3864\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3, 4, 6 and 12.
Its isogeny class 38640bi
consists of 8 curves linked by isogenies of
degrees dividing 12.
Twists
The minimal quadratic twist of this elliptic curve is 4830bc4, its twist by $-4$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{46}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-2}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-23}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-1}) \) | \(\Z/6\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-2}, \sqrt{-23})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(i, \sqrt{46})\) | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$4$ | \(\Q(\zeta_{8})\) | \(\Z/12\Z\) | Not in database |
$4$ | \(\Q(i, \sqrt{23})\) | \(\Z/12\Z\) | Not in database |
$6$ | 6.2.302228280000.1 | \(\Z/6\Z\) | Not in database |
$8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.0.21309228580864.28 | \(\Z/8\Z\) | Not in database |
$8$ | deg 8 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.18339659776.1 | \(\Z/2\Z \oplus \Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/12\Z\) | Not in database |
$12$ | deg 12 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/24\Z\) | Not in database |
$16$ | deg 16 | \(\Z/24\Z\) | Not in database |
$18$ | 18.0.22852660434578068506322018985168708505600000000.1 | \(\Z/18\Z\) | 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 | nonsplit | nonsplit | nonsplit | ss | ord | ord | ord | split | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | - | 1 | 1 | 3 | 1,1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 |
$\mu$-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
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.