Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-21389x+1210704\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-21389xz^2+1210704z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-342219x+77142854\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(98, 171\right)\) | \(\left(161, 1305\right)\) |
$\hat{h}(P)$ | ≈ | $0.28477848186012122418297453071$ | $0.35332646096412333997240794315$ |
Integral points
\( \left(-154, 990\right) \), \( \left(-154, -837\right) \), \( \left(-147, 1151\right) \), \( \left(-147, -1005\right) \), \( \left(-112, 1536\right) \), \( \left(-112, -1425\right) \), \( \left(-4, 1140\right) \), \( \left(-4, -1137\right) \), \( \left(62, 315\right) \), \( \left(62, -378\right) \), \( \left(77, 87\right) \), \( \left(77, -165\right) \), \( \left(84, -4\right) \), \( \left(84, -81\right) \), \( \left(98, 171\right) \), \( \left(98, -270\right) \), \( \left(108, 335\right) \), \( \left(108, -444\right) \), \( \left(161, 1305\right) \), \( \left(161, -1467\right) \), \( \left(392, 7080\right) \), \( \left(392, -7473\right) \), \( \left(449, 8829\right) \), \( \left(449, -9279\right) \), \( \left(854, 24174\right) \), \( \left(854, -25029\right) \), \( \left(81242, 23115627\right) \), \( \left(81242, -23196870\right) \)
Invariants
Conductor: | \( 20097 \) | = | $3^{2} \cdot 7 \cdot 11 \cdot 29$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-1418779930491 $ | = | $-1 \cdot 3^{7} \cdot 7^{5} \cdot 11^{3} \cdot 29 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{1484391946907017}{1946200179} \) | = | $-1 \cdot 3^{-1} \cdot 7^{-5} \cdot 11^{-3} \cdot 29^{-1} \cdot 114073^{3}$ | 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: | $1.2385633673350460073786429487\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $0.68925722300099116168102033024\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $0.9047161252289287\dots$ | |||
Szpiro ratio: | $4.191194264530338\dots$ |
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $0.084558559091557871573809650780\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.85101680223128406211966533986\dots$ | 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: | $ 60 $ = $ 2^{2}\cdot5\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: | $1$ | 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^{(2)}(E,1)/2! $ ≈ $ 4.3176452735629591259010875472 $ | 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
$\displaystyle 4.317645274 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.851017 \cdot 0.084559 \cdot 60}{1^2} \approx 4.317645274$
Modular invariants
Modular form 20097.2.a.a
For more coefficients, see the Downloads section to the right.
Modular degree: | 48000 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $4$ | $I_{1}^{*}$ | Additive | -1 | 2 | 7 | 1 |
$7$ | $5$ | $I_{5}$ | Split multiplicative | -1 | 1 | 5 | 5 |
$11$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$29$ | $1$ | $I_{1}$ | Non-split multiplicative | 1 | 1 | 1 | 1 |
Galois representations
The $\ell$-adic Galois representation has maximal image for all primes $\ell$.
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 13398 = 2 \cdot 3 \cdot 7 \cdot 11 \cdot 29 \), index $2$, genus $0$, and generators
$\left(\begin{array}{rr} 1 & 1 \\ 13397 & 0 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 2 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 6469 & 2 \\ 6469 & 3 \end{array}\right),\left(\begin{array}{rr} 13397 & 2 \\ 13396 & 3 \end{array}\right),\left(\begin{array}{rr} 8933 & 2 \\ 8933 & 3 \end{array}\right),\left(\begin{array}{rr} 5743 & 2 \\ 5743 & 3 \end{array}\right),\left(\begin{array}{rr} 8527 & 2 \\ 8527 & 3 \end{array}\right)$.
The torsion field $K:=\Q(E[13398])$ is a degree-$2613739290624000$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/13398\Z)$.
Isogenies
This curve has no rational isogenies. Its isogeny class 20097.a consists of this curve only.
Twists
The minimal quadratic twist of this elliptic curve is 6699.e1, its twist by $-3$.
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 $E(\Q)_{\rm tors}$ (which is trivial) are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.6699.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.300628350099.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | 8.2.300827790244107.2 | \(\Z/3\Z\) | Not in database |
$12$ | deg 12 | \(\Z/4\Z\) | 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 | add | ord | split | split | ord | ord | ord | ord | nonsplit | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 2 | - | 6 | 3 | 3 | 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 |
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.