Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2=x^3+x^2-2245x+83975\)
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(homogenize, simplify) |
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\(y^2z=x^3+x^2z-2245xz^2+83975z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-181872x+61763364\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{3}\Z\)
Infinite order Mordell-Weil generator and height
| $P$ | = |
\(\left(50, 315\right)\)
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| $\hat{h}(P)$ | ≈ | $0.23832444413787705089534664050$ |
Torsion generators
\( \left(5, 270\right) \)
Integral points
\((-55,\pm 210)\), \((-22,\pm 351)\), \((5,\pm 270)\), \((25,\pm 210)\), \((29,\pm 210)\), \((50,\pm 315)\), \((113,\pm 1134)\), \((365,\pm 6930)\), \((545,\pm 12690)\), \((10970,\pm 1149015)\)
Invariants
| Conductor: | \( 4620 \) | = | $2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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| Discriminant: | $-2376446688000 $ | = | $-1 \cdot 2^{8} \cdot 3^{9} \cdot 5^{3} \cdot 7^{3} \cdot 11 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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| j-invariant: | \( -\frac{4890195460096}{9282994875} \) | = | $-1 \cdot 2^{16} \cdot 3^{-9} \cdot 5^{-3} \cdot 7^{-3} \cdot 11^{-1} \cdot 421^{3}$ | 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: | $1.0645077817841732981821979203\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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| Stable Faltings height: | $0.60240966141087642523737650600\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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BSD invariants
| Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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| Regulator: | $0.23832444413787705089534664050\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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| Real period: | $0.72872311336521791304126503295\dots$ | 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: | $ 243 $ = $ 3\cdot3^{2}\cdot3\cdot3\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: | $3$ | 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: | $1$ (exact) | comment: Order of Sha
sage: E.sha().an_numerical()
magma: MordellWeilShaInformation(E);
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| Special value: | $ L'(E,1) $ ≈ $ 4.6891583349260954900707290932 $ | 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
$\displaystyle 4.689158335 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.728723 \cdot 0.238324 \cdot 243}{3^2} \approx 4.689158335$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 7776 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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| $ \Gamma_0(N) $-optimal: | yes | |
| Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
| prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
|---|---|---|---|---|---|---|---|
| $2$ | $3$ | $IV^{*}$ | Additive | -1 | 2 | 8 | 0 |
| $3$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
| $5$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
| $7$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
| $11$ | $1$ | $I_{1}$ | 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 |
|---|---|---|
| $3$ | 3B.1.1 | 3.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 2310 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \), index $16$, genus $0$, and generators
$\left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right),\left(\begin{array}{rr} 2305 & 6 \\ 2304 & 7 \end{array}\right),\left(\begin{array}{rr} 661 & 6 \\ 1983 & 19 \end{array}\right),\left(\begin{array}{rr} 1387 & 6 \\ 1851 & 19 \end{array}\right),\left(\begin{array}{rr} 211 & 6 \\ 633 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3 & 4 \\ 8 & 11 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 6 & 1 \end{array}\right),\left(\begin{array}{rr} 1926 & 391 \\ 1543 & 1944 \end{array}\right)$.
The torsion field $K:=\Q(E[2310])$ is a degree-$1108980$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/2310\Z)$.
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.
Iwasawa invariants
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Reduction type | add | split | split | split | nonsplit | ord | ord | ord | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | - | 6 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 1 | 3 | 1 | 1 | 3 | 1 |
| $\mu$-invariant(s) | - | 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.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 4620n
consists of 2 curves linked by isogenies of
degree 3.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{3}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $3$ | 3.1.4620.1 | \(\Z/6\Z\) | Not in database |
| $6$ | 6.0.24652782000.1 | \(\Z/2\Z \oplus \Z/6\Z\) | Not in database |
| $6$ | 6.0.6324912.1 | \(\Z/3\Z \oplus \Z/3\Z\) | Not in database |
| $9$ | 9.3.288178803000000.7 | \(\Z/9\Z\) | Not in database |
| $12$ | deg 12 | \(\Z/12\Z\) | Not in database |
| $18$ | 18.0.56280390065562432115862208000000.1 | \(\Z/3\Z \oplus \Z/6\Z\) | 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.