Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
\(y^2+xy+y=x^3-x^2-1112x-55492\) | (homogenize, simplify) |
\(y^2z+xyz+yz^2=x^3-x^2z-1112xz^2-55492z^3\) | (dehomogenize, simplify) |
\(y^2=x^3-17787x-3569258\) | (homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z\)
Infinite order Mordell-Weil generators and heights
$P$ | = | \(\left(58, 240\right)\) | \(\left(597, 14254\right)\) |
$\hat{h}(P)$ | ≈ | $0.90336827061961534309571528724$ | $1.7912477935888324968358801339$ |
Integral points
\( \left(48, -20\right) \), \( \left(48, -29\right) \), \( \left(58, 240\right) \), \( \left(58, -299\right) \), \( \left(102, 889\right) \), \( \left(102, -992\right) \), \( \left(157, 1824\right) \), \( \left(157, -1982\right) \), \( \left(597, 14254\right) \), \( \left(597, -14852\right) \), \( \left(405037, 257572984\right) \), \( \left(405037, -257978022\right) \)
Invariants
Conductor: | \( 53361 \) | = | $3^{2} \cdot 7^{2} \cdot 11^{2}$ | comment: Conductor
sage: E.conductor().factor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
oscar: conductor(E)
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Discriminant: | $-1255701777561 $ | = | $-1 \cdot 3^{6} \cdot 7^{6} \cdot 11^{4} $ | comment: Discriminant
sage: E.discriminant().factor()
gp: E.disc
magma: Discriminant(E);
oscar: discriminant(E)
|
j-invariant: | \( -121 \) | = | $-1 \cdot 11^{2}$ | 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: | $1.0043003807460324088653595323\dots$ | gp: ellheight(E)
magma: FaltingsHeight(E);
oscar: faltings_height(E)
|
||
Stable Faltings height: | $-1.3172592623818026040722539839\dots$ | magma: StableFaltingsHeight(E);
oscar: stable_faltings_height(E)
|
BSD invariants
Analytic rank: | $2$ | sage: E.analytic_rank()
gp: ellanalyticrank(E)
magma: AnalyticRank(E);
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Regulator: | $1.5470453485844442395067601164\dots$ | comment: Regulator
sage: E.regulator()
G = E.gen \\ if available
magma: Regulator(E);
|
Real period: | $0.36358524797052414255707168895\dots$ | 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);
|
Tamagawa product: | $ 12 $ = $ 2\cdot2\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)
|
Torsion order: | $1$ | comment: Torsion order
sage: E.torsion_order()
gp: elltors(E)[1]
magma: Order(TorsionSubgroup(E));
oscar: prod(torsion_structure(E)[1])
|
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! $ ≈ $ 6.7497944002406534353273963246 $ | comment: Special L-value
r = E.rank();
gp: [r,L1r] = ellanalyticrank(E); L1r/r!
magma: Lr1 where r,Lr1 := AnalyticRank(E: Precision:=12);
|
BSD formula
$\displaystyle 6.749794400 \approx L^{(2)}(E,1)/2! \overset{?}{=} \frac{\# Ш(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \approx \frac{1 \cdot 0.363585 \cdot 1.547045 \cdot 12}{1^2} \approx 6.749794400$
Modular invariants
Modular form 53361.2.a.r
For more coefficients, see the Downloads section to the right.
Modular degree: | 51840 | comment: Modular degree
sage: E.modular_degree()
gp: ellmoddegree(E)
magma: ModularDegree(E);
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$ \Gamma_0(N) $-optimal: | yes | |
Manin constant: | 1 | comment: Manin constant
magma: ManinConstant(E);
|
Local data
This elliptic curve is not semistable. There are 3 primes of bad reduction:
prime | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord($N$) | ord($\Delta$) | ord$(j)_{-}$ |
---|---|---|---|---|---|---|---|
$3$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$7$ | $2$ | $I_0^{*}$ | Additive | -1 | 2 | 6 | 0 |
$11$ | $3$ | $IV$ | Additive | -1 | 2 | 4 | 0 |
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 |
---|---|---|
$2$ | 2G | 4.2.0.1 |
$11$ | 11B.10.4 | 11.60.1.4 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 1848 = 2^{3} \cdot 3 \cdot 7 \cdot 11 \), index $480$, genus $16$, and generators
$\left(\begin{array}{rr} 263 & 0 \\ 0 & 1847 \end{array}\right),\left(\begin{array}{rr} 441 & 1408 \\ 440 & 441 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1408 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 924 & 1 \end{array}\right),\left(\begin{array}{rr} 1156 & 231 \\ 693 & 463 \end{array}\right),\left(\begin{array}{rr} 925 & 924 \\ 924 & 925 \end{array}\right),\left(\begin{array}{rr} 1231 & 0 \\ 0 & 1847 \end{array}\right),\left(\begin{array}{rr} 337 & 0 \\ 0 & 841 \end{array}\right),\left(\begin{array}{rr} 925 & 924 \\ 1386 & 1 \end{array}\right),\left(\begin{array}{rr} 925 & 462 \\ 462 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 924 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 925 & 1617 \\ 0 & 1387 \end{array}\right),\left(\begin{array}{rr} 1 & 84 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[1848])$ is a degree-$4087480320$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/1848\Z)$.
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
11.
Its isogeny class 53361bm
consists of 2 curves linked by isogenies of
degree 11.
Twists
The minimal quadratic twist of this elliptic curve is 121c1, its twist by $21$.
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 |
---|---|---|---|
$3$ | 3.1.484.1 | \(\Z/2\Z\) | Not in database |
$6$ | 6.0.937024.1 | \(\Z/2\Z \oplus \Z/2\Z\) | Not in database |
$8$ | 8.2.76879700667.4 | \(\Z/3\Z\) | Not in database |
$10$ | 10.0.9630096522760791.1 | \(\Z/11\Z\) | 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 | add | add | ord | ord | ord | ord | ord | ord | ord | ord | ss | ord |
$\lambda$-invariant(s) | ? | - | 2 | - | - | 2 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2,2 | 2 |
$\mu$-invariant(s) | ? | - | 0 | - | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0,0 | 0 |
An entry ? indicates that the invariants have not yet been computed.
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.