Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy=x^3+12948x+421776\) | (homogenize, simplify) |
\(y^2z+xyz=x^3+12948xz^2+421776z^3\) | (dehomogenize, simplify) |
\(y^2=x^3+16780581x+19628039286\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{9}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(-30, 96\right)\) |
$\hat{h}(P)$ | ≈ | $1.7357307630471408735001735385$ |
Torsion generators
\( \left(-24, 324\right) \)
Integral points
\( \left(-30, 96\right) \), \( \left(-30, -66\right) \), \( \left(-24, 324\right) \), \( \left(-24, -300\right) \), \( \left(24, 852\right) \), \( \left(24, -876\right) \), \( \left(54, 1104\right) \), \( \left(54, -1158\right) \), \( \left(80, 1364\right) \), \( \left(80, -1444\right) \), \( \left(132, 2040\right) \), \( \left(132, -2172\right) \), \( \left(216, 3540\right) \), \( \left(216, -3756\right) \), \( \left(312, 5748\right) \), \( \left(312, -6060\right) \), \( \left(600, 14676\right) \), \( \left(600, -15276\right) \), \( \left(1536, 59604\right) \), \( \left(1536, -61140\right) \), \( \left(4344, 284244\right) \), \( \left(4344, -288588\right) \), \( \left(9372, 902688\right) \), \( \left(9372, -912060\right) \)
Invariants
Conductor: | \( 1482 \) | = | $2 \cdot 3 \cdot 13 \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-215384711233536 $ | = | $-1 \cdot 2^{18} \cdot 3^{9} \cdot 13^{3} \cdot 19 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( \frac{240065464752030527}{215384711233536} \) | = | $2^{-18} \cdot 3^{-9} \cdot 13^{-3} \cdot 17^{3} \cdot 19^{-1} \cdot 36559^{3}$ | 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: | $1.4376715021532606170302076599\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.4376715021532606170302076599\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0026239435204949\dots$ | |||
Szpiro ratio: | $5.48128720511031\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.7357307630471408735001735385\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.36604694470320968107915360838\dots$ | 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: | $ 486 $ = $ ( 2 \cdot 3^{2} )\cdot3^{2}\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: | $9$ | 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'(E,1) $ ≈ $ 3.8121536558446603260233098916 $ | 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
$\displaystyle 3.812153656 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.366047 \cdot 1.735731 \cdot 486}{9^2} \approx 3.812153656$
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) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is semistable. There are 4 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $18$ | $I_{18}$ | Split multiplicative | -1 | 1 | 18 | 18 |
$3$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
$13$ | $3$ | $I_{3}$ | Split multiplicative | -1 | 1 | 3 | 3 |
$19$ | $1$ | $I_{1}$ | 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 | 9.72.0.5 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 8892 = 2^{2} \cdot 3^{2} \cdot 13 \cdot 19 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 8875 & 18 \\ 8874 & 19 \end{array}\right),\left(\begin{array}{rr} 4457 & 18 \\ 4779 & 6967 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 4447 & 18 \\ 4455 & 163 \end{array}\right),\left(\begin{array}{rr} 3421 & 18 \\ 4113 & 163 \end{array}\right),\left(\begin{array}{rr} 3745 & 18 \\ 7029 & 163 \end{array}\right)$.
The torsion field $K:=\Q(E[8892])$ is a degree-$8363681464320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/8892\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 1482.j
consists of 3 curves linked by isogenies of
degrees dividing 9.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{9}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.2964.1 | \(\Z/18\Z\) | Not in database |
$6$ | 6.0.26039617344.2 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
$6$ | 6.0.3518667.2 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$9$ | 9.3.577661352086139003072.4 | \(\Z/27\Z\) | Not in database |
$12$ | deg 12 | \(\Z/36\Z\) | Not in database |
$18$ | 18.0.310929384841571919202621092605952.1 | \(\Z/3\Z \oplus \Z/18\Z\) | 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 | split | split | ord | ord | ord | split | ss | split | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 3 | 2 | 1 | 1 | 1 | 2 | 1,1 | 2 | 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 |
$p$-adic regulators
Note: $p$-adic regulator data only exists for primes $p\ge 5$ of good ordinary reduction.