Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3+x^2+315687x+36481656\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3+x^2z+315687xz^2+36481656z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+409130325x+1695951195750\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(670, 23102\right)\) | \(\left(40, 6992\right)\) |
$\hat{h}(P)$ | ≈ | $1.6003148592128997038707097294$ | $4.3128898096583381634412951410$ |
Torsion generators
\( \left(-\frac{445}{4}, \frac{441}{8}\right) \)
Integral points
\( \left(-81, 3263\right) \), \( \left(-81, -3183\right) \), \( \left(40, 6992\right) \), \( \left(40, -7033\right) \), \( \left(340, 13367\right) \), \( \left(340, -13708\right) \), \( \left(670, 23102\right) \), \( \left(670, -23773\right) \), \( \left(4311, 283395\right) \), \( \left(4311, -287707\right) \)
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: | $-2584307427978515625 $ | = | $-1 \cdot 3^{2} \cdot 5^{14} \cdot 19^{6} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{4733169839}{3515625} \) | = | $3^{-2} \cdot 5^{-8} \cdot 23^{3} \cdot 73^{3}$ | 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: | $2.2212344639973992845699129461\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $-0.055703981802871132734980436459\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0558519748642754\dots$ | |||
Szpiro ratio: | $4.859821994583027\dots$ |
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: | $0.16377099645915117351187995924\dots$ | 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: | $ 16 $ = $ 2\cdot2^{2}\cdot2 $ | 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
$\displaystyle 4.492241598 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.163771 \cdot 6.857505 \cdot 16}{2^2} \approx 4.492241598$
Modular invariants
Modular form 27075.2.a.f
For more coefficients, see the Downloads section to the right.
Modular degree: | 331776 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_{2}$ | Non-split multiplicative | 1 | 1 | 2 | 2 |
$5$ | $4$ | $I_{8}^{*}$ | Additive | 1 | 2 | 14 | 8 |
$19$ | $2$ | $I_0^{*}$ | 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 | 8.48.0.197 |
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
$\left(\begin{array}{rr} 1 & 0 \\ 32 & 1 \end{array}\right),\left(\begin{array}{rr} 25 & 16 \\ 7384 & 8009 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 4 & 129 \end{array}\right),\left(\begin{array}{rr} 2585 & 2432 \\ 6422 & 2889 \end{array}\right),\left(\begin{array}{rr} 1823 & 6232 \\ 0 & 9119 \end{array}\right),\left(\begin{array}{rr} 9089 & 32 \\ 9088 & 33 \end{array}\right),\left(\begin{array}{rr} 7199 & 0 \\ 0 & 9119 \end{array}\right),\left(\begin{array}{rr} 1 & 32 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6727 & 5776 \\ 855 & 3193 \end{array}\right),\left(\begin{array}{rr} 4333 & 5776 \\ 5054 & 5625 \end{array}\right)$.
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 and 8.
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 15a4, 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 |
---|---|---|---|
$2$ | \(\Q(\sqrt{-1}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$2$ | \(\Q(\sqrt{95}) \) | \(\Z/4\Z\) | Not in database |
$2$ | \(\Q(\sqrt{-95}) \) | \(\Z/8\Z\) | Not in database |
$4$ | \(\Q(i, \sqrt{95})\) | \(\Z/2\Z \oplus \Z/8\Z\) | Not in database |
$4$ | \(\Q(\sqrt{6}, \sqrt{95})\) | \(\Z/8\Z\) | Not in database |
$4$ | \(\Q(\sqrt{-6}, \sqrt{95})\) | \(\Z/8\Z\) | Not in database |
$8$ | 8.0.432373800960000.267 | \(\Z/4\Z \oplus \Z/8\Z\) | Not in database |
$8$ | 8.0.1688960160000.69 | \(\Z/16\Z\) | Not in database |
$8$ | deg 8 | \(\Z/6\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/2\Z \oplus \Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/16\Z\) | Not in database |
$16$ | deg 16 | \(\Z/16\Z\) | 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 | \(\Z/24\Z\) | 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) | 1 | 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.