Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2-188x-159844\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z-188xz^2-159844z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-243675x-7454018250\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(131, 1378\right)\) | \(\left(80, 547\right)\) |
$\hat{h}(P)$ | ≈ | $2.1812939857204240064565331279$ | $3.1674976105707928414269053822$ |
Torsion generators
\( \left(55, -28\right) \)
Integral points
\( \left(55, -28\right) \), \( \left(80, 547\right) \), \( \left(80, -628\right) \), \( \left(131, 1378\right) \), \( \left(131, -1510\right) \), \( \left(416, 8275\right) \), \( \left(416, -8692\right) \)
Invariants
Conductor: | \( 27075 \) | = | $3 \cdot 5^{2} \cdot 19^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-11026378359375 $ | = | $-1 \cdot 3 \cdot 5^{7} \cdot 19^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{1}{15} \) | = | $-1 \cdot 3^{-1} \cdot 5^{-1}$ | 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: | $1.1815136931574813204440647639\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-1.0954247526427890968608286187\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.1980768440515948\dots$ | |||
Szpiro ratio: | $3.672812616714911\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $6.8575048310480019863121788050\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.32754199291830234702375991848\dots$ | 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: | $ 8 $ = $ 1\cdot2\cdot2^{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: | $2$ | 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^{(2)}(E,1)/2! $ ≈ $ 4.4922415976166975985896004582 $ | 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
$\displaystyle 4.492241598 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.327542 \cdot 6.857505 \cdot 8}{2^2} \approx 4.492241598$
Modular invariants
Modular form 27075.2.a.f
For more coefficients, see the Downloads section to the right.
Modular degree: | 41472 | 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)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
$5$ | $2$ | $I_{1}^{*}$ | Additive | 1 | 2 | 7 | 1 |
$19$ | $4$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
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 | 32.48.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9120 = 2^{5} \cdot 3 \cdot 5 \cdot 19 \), index $768$, genus $13$, and generators
$\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 6423 & 5282 \\ 5434 & 7183 \end{array}\right),\left(\begin{array}{rr} 23 & 18 \\ 6558 & 7115 \end{array}\right),\left(\begin{array}{rr} 7466 & 5757 \\ 7315 & 1652 \end{array}\right),\left(\begin{array}{rr} 1312 & 8189 \\ 4427 & 1122 \end{array}\right),\left(\begin{array}{rr} 9089 & 32 \\ 9088 & 33 \end{array}\right),\left(\begin{array}{rr} 5 & 28 \\ 68 & 381 \end{array}\right),\left(\begin{array}{rr} 7199 & 0 \\ 0 & 9119 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2432 \\ 570 & 6271 \end{array}\right)$.
The torsion field $K:=\Q(E[9120])$ is a degree-$1452382617600$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9120\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 4, 8 and 16.
Its isogeny class 27075g
consists of 8 curves linked by isogenies of
degrees dividing 16.
Twists
The minimal quadratic twist of this elliptic curve is 15a8, its twist by $-95$.
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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{57}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-95}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-15}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-15}, \sqrt{57})\) | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-3}, \sqrt{-95})\) | \(\Z/8\Z\) | Not in database |
$4$ | \(\Q(\sqrt{5}, \sqrt{-19})\) | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.380016036000000.22 | \(\Z/2\Z \oplus \Z/4\Z\) | Not in database |
$8$ | 8.4.380016036000000.15 | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.6597500625.1 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.2036265625.1 | \(\Z/16\Z\) | Not in database |
$8$ | 8.0.164937515625.1 | \(\Z/16\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$16$ | deg 16 | \(\Z/16\Z\) | Not in database |
$16$ | 16.0.27204384060547119140625.1 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/32\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | Not in database |
$16$ | deg 16 | \(\Z/12\Z\) | 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 | ord | nonsplit | add | ss | ord | ord | ord | add | ss | ord | ss | ord | ord | ord | ord |
$\lambda$-invariant(s) | 3 | 2 | - | 2,2 | 2 | 2 | 2 | - | 2,2 | 2 | 2,2 | 2 | 2 | 2 | 2 |
$\mu$-invariant(s) | 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.