Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+y=x^3+x^2-2564593x-1581651042\) | (homogenize, simplify) |
\(y^2z+yz^2=x^3+x^2z-2564593xz^2-1581651042z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-3323712960x-73753626450096\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-\frac{4028939}{4356}, -\frac{145117}{287496}\right)\) |
$\hat{h}(P)$ | ≈ | $5.5276396655685530077329627716$ |
Integral points
None
Invariants
Conductor: | \( 1369 \) | = | $37^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $94931877133 $ | = | $37^{7} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{727057727488000}{37} \) | = | $2^{18} \cdot 5^{3} \cdot 37^{-1} \cdot 281^{3}$ | 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);
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Sato-Tate group: | $\mathrm{SU}(2)$ | |||
Faltings height: | $2.0275405158458155771908328410\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.22208155952370335500678500549\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0859831071145252\dots$ | |||
Szpiro ratio: | $7.738411094247738\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $5.5276396655685530077329627716\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.11930123168897977776631018948\dots$ | 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);
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Tamagawa product: | $ 4 $ = $ 2^{2} $ | 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: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
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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) $ ≈ $ 2.6378168817407545482678312276 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
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BSD formula
$\displaystyle 2.637816882 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.119301 \cdot 5.527640 \cdot 4}{1^2} \approx 2.637816882$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 8208 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There is only one prime of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$37$ | $4$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 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 |
---|---|---|
$3$ | 3B | 27.36.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1998 = 2 \cdot 3^{3} \cdot 37 \), index $1296$, genus $43$, and generators
$\left(\begin{array}{rr} 31 & 36 \\ 172 & 1231 \end{array}\right),\left(\begin{array}{rr} 1801 & 129 \\ 1491 & 1226 \end{array}\right),\left(\begin{array}{rr} 1945 & 54 \\ 1944 & 55 \end{array}\right),\left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 28 & 27 \\ 729 & 703 \end{array}\right),\left(\begin{array}{rr} 1226 & 1953 \\ 1667 & 1418 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 54 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1998])$ is a degree-$2656732608$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1998\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 1369.c
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 37.b1, its twist by $37$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-111}) \) | \(\Z/3\Z\) | Not in database |
$3$ | 3.3.148.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.6.810448.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.2.624095613.1 | \(\Z/3\Z\) | Not in database |
$6$ | 6.0.1364897105631.2 | \(\Z/9\Z\) | Not in database |
$6$ | 6.0.21882096.1 | \(\Z/6\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | 12.0.3505458007492611921.2 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
$12$ | 12.0.1360806507640526787369.3 | \(\Z/9\Z\) | Not in database |
$12$ | 12.0.478826125353216.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.6.995665220963833277921326682112.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$18$ | 18.0.10415009883224736175862338741239622004736.1 | \(\Z/18\Z\) | 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 | ss | ord | ss | ord | ord | ord | ord | ord | ord | ord | ord | add | ord | ord | ord |
$\lambda$-invariant(s) | ? | 3 | 1,3 | 1 | 1 | 1 | 3 | 1 | 3 | 1 | 1 | - | 1 | 1 | 1 |
$\mu$-invariant(s) | ? | 2 | 0,0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | - | 0 | 0 | 0 |
An entry ? indicates that the invariants have not yet been computed.
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.