Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3-x^2-982409x-374543312\) | (homogenize, simplify) |
\(y^2z+xyz=x^3-x^2z-982409xz^2-374543312z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-15718547x-23986490514\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
$P$ | $\hat{h}(P)$ | Order |
---|---|---|
$(13348/9, 1000658/27)$ | $6.6552718759833089389679662935$ | $\infty$ |
$(-572, 286)$ | $0$ | $2$ |
Integral points
\( \left(-572, 286\right) \)
Invariants
Conductor: | $N$ | = | \( 1045 \) | = | $5 \cdot 11 \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $\Delta$ | = | $47155625$ | = | $5^{4} \cdot 11 \cdot 19^{3} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | $j$ | = | \( \frac{104857852278310619039721}{47155625} \) | = | $3^{3} \cdot 5^{-4} \cdot 11^{-1} \cdot 13^{3} \cdot 19^{-3} \cdot 313^{3} \cdot 3863^{3}$ | comment: j-invariant
sage: E.j_invariant().factor()
gp: E.j
magma: jInvariant(E);
oscar: j_invariant(E)
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Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) | sage: E.has_cm()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $1.7158240888248362330886097080$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.7158240888248362330886097080$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $Q$ | ≈ | $1.0482504438817541$ | |||
Szpiro ratio: | $\sigma_{m}$ | ≈ | $7.624946753788142$ |
BSD invariants
Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Mordell-Weil rank: | $r$ | = | $ 1$ | comment: Rank
sage: E.rank()
gp: [lower,upper] = ellrank(E)
magma: Rank(E);
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Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $6.6552718759833089389679662935$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $\Omega$ | ≈ | $0.15164442574914819669892743864$ | 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: | $\prod_{p}c_p$ | = | $ 12 $ = $ 2^{2}\cdot1\cdot3 $ | 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: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ | 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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Special value: | $ L'(E,1)$ | ≈ | $3.0277046455138353542434232564 $ | 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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Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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BSD formula
$\displaystyle 3.027704646 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.151644 \cdot 6.655272 \cdot 12}{2^2} \approx 3.027704646$
Modular invariants
For more coefficients, see the Downloads section to the right.
Modular degree: | 4416 | 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 at primes of bad reduction
This elliptic curve is semistable. There are 3 primes $p$ of bad reduction:
$p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
---|---|---|---|---|---|---|---|
$5$ | $4$ | $I_{4}$ | split multiplicative | -1 | 1 | 4 | 4 |
$11$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
$19$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
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 | 8.12.0.6 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1672 = 2^{3} \cdot 11 \cdot 19 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 1665 & 8 \\ 1664 & 9 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 8 & 1 \end{array}\right),\left(\begin{array}{rr} 7 & 6 \\ 1666 & 1667 \end{array}\right),\left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 4 & 17 \end{array}\right),\left(\begin{array}{rr} 217 & 212 \\ 214 & 1047 \end{array}\right),\left(\begin{array}{rr} 1372 & 1 \\ 783 & 6 \end{array}\right),\left(\begin{array}{rr} 1467 & 1466 \\ 1058 & 219 \end{array}\right),\left(\begin{array}{rr} 624 & 3 \\ 181 & 2 \end{array}\right)$.
The torsion field $K:=\Q(E[1672])$ is a degree-$52005888000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1672\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
$\ell$ | Reduction type | Serre weight | Serre conductor |
---|---|---|---|
$2$ | good | $2$ | \( 209 = 11 \cdot 19 \) |
$3$ | good | $2$ | \( 55 = 5 \cdot 11 \) |
$5$ | split multiplicative | $6$ | \( 209 = 11 \cdot 19 \) |
$11$ | nonsplit multiplicative | $12$ | \( 95 = 5 \cdot 19 \) |
$19$ | split multiplicative | $20$ | \( 55 = 5 \cdot 11 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2 and 4.
Its isogeny class 1045b
consists of 4 curves linked by isogenies of
degrees dividing 4.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$2$ | \(\Q(\sqrt{209}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
$2$ | \(\Q(\sqrt{-19}) \) | \(\Z/4\Z\) | not in database |
$2$ | \(\Q(\sqrt{-11}) \) | \(\Z/4\Z\) | not in database |
$4$ | \(\Q(\sqrt{-11}, \sqrt{-19})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$4$ | 4.0.75449.1 | \(\Z/8\Z\) | not in database |
$8$ | 8.4.21336229885501696.2 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
$8$ | 8.0.163720581376.3 | \(\Z/8\Z\) | not in database |
$8$ | 8.0.688798743721.1 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
$8$ | 8.2.20012416875.1 | \(\Z/6\Z\) | 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 |
$16$ | 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 | ss | split | ss | nonsplit | ord | ord | split | ord | ord | ord | ord | ord | ord | ss |
$\lambda$-invariant(s) | 1 | 3,3 | 2 | 3,1 | 1 | 3 | 1 | 2 | 1 | 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 | 0,0 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.