Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2=x^3-x^2-136641x+18524097\) | (homogenize, simplify) |
\(y^2z=x^3-x^2z-136641xz^2+18524097z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-11067948x+13470862896\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(49, 3456\right)\) |
$\hat{h}(P)$ | ≈ | $2.8371610541089729837953252274$ |
Integral points
\((49,\pm 3456)\)
Invariants
Conductor: | \( 25152 \) | = | $2^{6} \cdot 3 \cdot 131$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $15768123642740736 $ | = | $2^{23} \cdot 3^{15} \cdot 131 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{1076291879750641}{60150618144} \) | = | $2^{-5} \cdot 3^{-15} \cdot 131^{-1} \cdot 102481^{3}$ | 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: | $1.8630775765472671002781430561\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.82335680570734913615229487391\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9761007764835719\dots$ | |||
Szpiro ratio: | $4.647229394001553\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $2.8371610541089729837953252274\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.38663094586195378235152117017\dots$ | 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}\cdot1\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: | $1$ | 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) $ ≈ $ 4.3877370476514002377356574259 $ | 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
$\displaystyle 4.387737048 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.386631 \cdot 2.837161 \cdot 4}{1^2} \approx 4.387737048$
Modular invariants
Modular form 25152.2.a.j
For more coefficients, see the Downloads section to the right.
Modular degree: | 161280 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $4$ | $I_{13}^{*}$ | Additive | 1 | 6 | 23 | 5 |
$3$ | $1$ | $I_{15}$ | Non-split multiplicative | 1 | 1 | 15 | 15 |
$131$ | $1$ | $I_{1}$ | 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 |
---|---|---|
$5$ | 5B.4.1 | 5.12.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 15720 = 2^{3} \cdot 3 \cdot 5 \cdot 131 \), index $48$, genus $1$, and generators
$\left(\begin{array}{rr} 11791 & 10 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6 & 13 \\ 15665 & 15601 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 10 & 1 \end{array}\right),\left(\begin{array}{rr} 15714 & 15715 \\ 7865 & 4 \end{array}\right),\left(\begin{array}{rr} 6 & 5 \\ 10475 & 15716 \end{array}\right),\left(\begin{array}{rr} 11791 & 7870 \\ 0 & 8647 \end{array}\right),\left(\begin{array}{rr} 7086 & 5 \\ 7195 & 15716 \end{array}\right),\left(\begin{array}{rr} 15711 & 10 \\ 15710 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 10 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[15720])$ is a degree-$215458873344000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15720\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
5.
Its isogeny class 25152.j
consists of 2 curves linked by isogenies of
degree 5.
Twists
The minimal quadratic twist of this elliptic curve is 786.m2, its twist by $8$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{2}) \) | \(\Z/5\Z\) | Not in database |
$3$ | 3.3.3144.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.6.31077609984.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$6$ | 6.6.79077888.1 | \(\Z/10\Z\) | Not in database |
$8$ | deg 8 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | Not in database |
$12$ | deg 12 | \(\Z/2\Z \oplus \Z/10\Z\) | Not in database |
$16$ | deg 16 | \(\Z/15\Z\) | Not in database |
$20$ | 20.0.246485096653626481061419420776860254208000000000000000.1 | \(\Z/5\Z\) | 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 | 131 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Reduction type | add | nonsplit | ord | ord | ord | ord | ord | ss | ord | ss | ord | ord | ord | ord | ord | nonsplit |
$\lambda$-invariant(s) | - | 1 | 1 | 1 | 1 | 1 | 1 | 1,1 | 1 | 3,1 | 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 |
An entry - indicates that the invariants are not computed because the reduction is additive.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.