Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-8462840x-15758963160\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-8462840xz^2-15758963160z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-10967840667x-735217281670986\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(\frac{243618}{49}, \frac{86641458}{343}\right)\) |
$\hat{h}(P)$ | ≈ | $12.263533475735357376126728701$ |
Integral points
None
Invariants
Conductor: | \( 27690 \) | = | $2 \cdot 3 \cdot 5 \cdot 13 \cdot 71$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-68484636225100842737640 $ | = | $-1 \cdot 2^{3} \cdot 3 \cdot 5 \cdot 13^{7} \cdot 71^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( -\frac{67030445471226692469644161}{68484636225100842737640} \) | = | $-1 \cdot 2^{-3} \cdot 3^{-1} \cdot 5^{-1} \cdot 7^{6} \cdot 13^{-7} \cdot 71^{-7} \cdot 8290129^{3}$ | 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: | $3.0772210886679891993291478289\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
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Stable Faltings height: | $3.0772210886679891993291478289\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
|
Regulator: | $12.263533475735357376126728701\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.042515992916836070217919259801\dots$ | 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: | $ 21 $ = $ 3\cdot1\cdot1\cdot1\cdot7 $ | 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) $ ≈ $ 10.949322350184676231192661398 $ | 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
$\displaystyle 10.949322350 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.042516 \cdot 12.263533 \cdot 21}{1^2} \approx 10.949322350$
Modular invariants
Modular form 27690.2.a.bg
For more coefficients, see the Downloads section to the right.
Modular degree: | 3687936 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$3$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$5$ | $1$ | $I_{1}$ | Split multiplicative | -1 | 1 | 1 | 1 |
$13$ | $1$ | $I_{7}$ | Non-split multiplicative | 1 | 1 | 7 | 7 |
$71$ | $7$ | $I_{7}$ | Split multiplicative | -1 | 1 | 7 | 7 |
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 |
---|---|---|
$7$ | 7B.1.3 | 7.48.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 775320 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 71 \), index $96$, genus $2$, and generators
$\left(\begin{array}{rr} 387661 & 14 \\ 387667 & 99 \end{array}\right),\left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 178921 & 14 \\ 477127 & 99 \end{array}\right),\left(\begin{array}{rr} 516881 & 14 \\ 516887 & 99 \end{array}\right),\left(\begin{array}{rr} 620257 & 14 \\ 465199 & 99 \end{array}\right),\left(\begin{array}{rr} 193831 & 14 \\ 581497 & 99 \end{array}\right),\left(\begin{array}{rr} 193833 & 553808 \\ 387646 & 636833 \end{array}\right),\left(\begin{array}{rr} 775307 & 14 \\ 775306 & 15 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 14 & 1 \end{array}\right),\left(\begin{array}{rr} 8 & 5 \\ 91 & 57 \end{array}\right),\left(\begin{array}{rr} 327601 & 14 \\ 742567 & 99 \end{array}\right)$.
The torsion field $K:=\Q(E[775320])$ is a degree-$487880870955319296000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/775320\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
7.
Its isogeny class 27690bc
consists of 2 curves linked by isogenies of
degree 7.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.110760.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.1358779046976000.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | \(\Q(\zeta_{7})\) | \(\Z/7\Z\) | Not in database |
$7$ | 7.1.12252303000000.10 | \(\Z/7\Z\) | Not in database |
$8$ | deg 8 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$18$ | 18.0.8765330231241754661514492110429319168000000.2 | \(\Z/14\Z\) | Not in database |
$21$ | 21.1.515948419432548652795384693815714046409770106880000000000000000000.1 | \(\Z/14\Z\) | 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 | 71 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | split | split | split | ord | ord | nonsplit | ord | ord | ord | ord | ord | ord | ss | ord | ord | split |
$\lambda$-invariant(s) | 9 | 2 | 2 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 1 | 2 |
$\mu$-invariant(s) | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 | 0 | 0 |
$p$-adic regulators
$p$-adic regulators are not yet computed for curves that are not $\Gamma_0$-optimal.