Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-3606647x+2637610319\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-3606647xz^2+2637610319z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-57706347x+168749354086\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{9}\Z\)
Infinite order Mordell-Weil generator and height
$P$ | = | \(\left(1077, 736\right)\) |
$\hat{h}(P)$ | ≈ | $1.7981775556969207378889408412$ |
Torsion generators
\( \left(1107, 286\right) \)
Integral points
\( \left(-1973, 46486\right) \), \( \left(-1973, -44514\right) \), \( \left(-573, 67486\right) \), \( \left(-573, -66914\right) \), \( \left(477, 31786\right) \), \( \left(477, -32264\right) \), \( \left(867, 12286\right) \), \( \left(867, -13154\right) \), \( \left(1027, 3486\right) \), \( \left(1027, -4514\right) \), \( \left(1077, 736\right) \), \( \left(1077, -1814\right) \), \( \left(1107, 286\right) \), \( \left(1107, -1394\right) \), \( \left(1219, 6558\right) \), \( \left(1219, -7778\right) \), \( \left(1527, 25486\right) \), \( \left(1527, -27014\right) \), \( \left(2787, 117886\right) \), \( \left(2787, -120674\right) \), \( \left(9027, 835486\right) \), \( \left(9027, -844514\right) \)
Invariants
Conductor: | \( 35910 \) | = | $2 \cdot 3^{3} \cdot 5 \cdot 7 \cdot 19$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-810819072000000000 $ | = | $-1 \cdot 2^{18} \cdot 3^{5} \cdot 5^{9} \cdot 7^{3} \cdot 19 $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
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j-invariant: | \( -\frac{21351492553819653671547}{3336704000000000} \) | = | $-1 \cdot 2^{-18} \cdot 3 \cdot 5^{-9} \cdot 7^{-3} \cdot 17^{3} \cdot 19^{-1} \cdot 37^{3} \cdot 53^{3} \cdot 577^{3}$ | 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: | $2.4475130041056643141624112166\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
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Stable Faltings height: | $1.9897578838272852760810590345\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
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$abc$ quality: | $1.0018860372141205\dots$ | |||
Szpiro ratio: | $5.42568233930996\dots$ |
BSD invariants
Analytic rank: | $1$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.7981775556969207378889408412\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
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Real period: | $0.27324839792906409646767394076\dots$ | 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: | $ 1458 $ = $ ( 2 \cdot 3^{2} )\cdot3\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) $ ≈ $ 8.8442844531513122796519662619 $ | 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
$\displaystyle 8.844284453 \approx L'(E,1) = \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.273248 \cdot 1.798178 \cdot 1458}{9^2} \approx 8.844284453$
Modular invariants
Modular form 35910.2.a.by
For more coefficients, see the Downloads section to the right.
Modular degree: | 1259712 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | no | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
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Local data
This elliptic curve is not semistable. There are 5 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$2$ | $18$ | $I_{18}$ | Split multiplicative | -1 | 1 | 18 | 18 |
$3$ | $3$ | $IV$ | Additive | 1 | 3 | 5 | 0 |
$5$ | $9$ | $I_{9}$ | Split multiplicative | -1 | 1 | 9 | 9 |
$7$ | $3$ | $I_{3}$ | 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 \( 11970 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 19 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 10090 & 9 \\ 10071 & 11962 \end{array}\right),\left(\begin{array}{rr} 2 & 9 \\ 11925 & 2458 \end{array}\right),\left(\begin{array}{rr} 11953 & 18 \\ 11952 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 10 & 181 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 81 & 73 \end{array}\right),\left(\begin{array}{rr} 10 & 9 \\ 9567 & 11962 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 5140 & 9 \\ 8541 & 11962 \end{array}\right)$.
The torsion field $K:=\Q(E[11970])$ is a degree-$19300803379200$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/11970\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3 and 9.
Its isogeny class 35910bt
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 35910v3, 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 $E(\Q)_{\rm tors}$ $\cong \Z/{9}\Z$ are as follows:
$[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
---|---|---|---|
$3$ | 3.1.17955.1 | \(\Z/18\Z\) | Not in database |
$6$ | 6.0.643152139875.1 | \(\Z/2\Z \oplus \Z/18\Z\) | Not in database |
$6$ | 6.0.23085974187.5 | \(\Z/3\Z \oplus \Z/9\Z\) | Not in database |
$9$ | 9.3.17819950099726067643000000.3 | \(\Z/27\Z\) | Not in database |
$12$ | deg 12 | \(\Z/36\Z\) | Not in database |
$18$ | 18.0.661370933225890561201372149193114200281671875.1 | \(\Z/3\Z \oplus \Z/18\Z\) | 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 | add | split | split | ord | ord | ss | split | ord | ord | ord | ord | ord | ord | ord |
$\lambda$-invariant(s) | 3 | - | 2 | 2 | 1 | 1 | 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 |
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.