Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3+21565828x+33710773538\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3+21565828xz^2+33710773538z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3+27949313709x+1572726002259438\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(38964, 7728373)$ | $2.4000084657602672263044698042$ | $\infty$ |
Integral points
\( \left(38964, 7728373\right) \), \( \left(38964, -7767338\right) \)
Invariants
| Conductor: | $N$ | = | \( 57498 \) | = | $2 \cdot 3 \cdot 7 \cdot 37^{2}$ |
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| Discriminant: | $\Delta$ | = | $-1132795794108988323231624$ | = | $-1 \cdot 2^{3} \cdot 3^{3} \cdot 7^{9} \cdot 37^{9} $ |
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| j-invariant: | $j$ | = | \( \frac{432326451325256207}{441510751160136} \) | = | $2^{-3} \cdot 3^{-3} \cdot 7^{-9} \cdot 17^{3} \cdot 19^{3} \cdot 37^{-3} \cdot 2341^{3}$ |
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| Endomorphism ring: | $\mathrm{End}(E)$ | = | $\Z$ | |||
| Geometric endomorphism ring: | $\mathrm{End}(E_{\overline{\Q}})$ | = | \(\Z\) (no potential complex multiplication) |
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| Sato-Tate group: | $\mathrm{ST}(E)$ | = | $\mathrm{SU}(2)$ | |||
| Faltings height: | $h_{\mathrm{Faltings}}$ | ≈ | $3.3023999520510117923011870134$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $1.4969409957288995701171391779$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0029599998384076$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $5.682141878448712$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $2.4000084657602672263044698042$ |
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| Real period: | $\Omega$ | ≈ | $0.057368200715011861503694255557$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 54 $ = $ 1\cdot3\cdot3^{2}\cdot2 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L'(E,1)$ | ≈ | $7.4349450385989848830775166181 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.434945039 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.057368 \cdot 2.400008 \cdot 54}{1^2} \\ & \approx 7.434945039\end{aligned}$$
Modular invariants
Modular form 57498.2.a.m
For more coefficients, see the Downloads section to the right.
| Modular degree: | 15956352 |
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| $ \Gamma_0(N) $-optimal: | no | |
| Manin constant: | 1 |
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Local data at primes of bad reduction
This elliptic curve is not semistable. There are 4 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $1$ | $I_{3}$ | nonsplit multiplicative | 1 | 1 | 3 | 3 |
| $3$ | $3$ | $I_{3}$ | split multiplicative | -1 | 1 | 3 | 3 |
| $7$ | $9$ | $I_{9}$ | split multiplicative | -1 | 1 | 9 | 9 |
| $37$ | $2$ | $I_{3}^{*}$ | additive | 1 | 2 | 9 | 3 |
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 |
|---|---|---|
| $3$ | 3Cs | 9.36.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 18648 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 37 \), index $144$, genus $3$, and generators
$\left(\begin{array}{rr} 10 & 9 \\ 9315 & 18640 \end{array}\right),\left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 9 \\ 9 & 82 \end{array}\right),\left(\begin{array}{rr} 13330 & 9 \\ 15975 & 18640 \end{array}\right),\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 9334 & 13995 \\ 4653 & 10352 \end{array}\right),\left(\begin{array}{rr} 18631 & 18 \\ 18630 & 19 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 18 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 3014 & 18639 \\ 1017 & 8 \end{array}\right),\left(\begin{array}{rr} 9334 & 9 \\ 13977 & 18640 \end{array}\right)$.
The torsion field $K:=\Q(E[18648])$ is a degree-$152347674673152$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/18648\Z)$.
The table below list all primes $\ell$ for which the Serre invariants associated to the mod-$\ell$ Galois representation are exceptional.
| $\ell$ | Reduction type | Serre weight | Serre conductor |
|---|---|---|---|
| $2$ | nonsplit multiplicative | $4$ | \( 28749 = 3 \cdot 7 \cdot 37^{2} \) |
| $3$ | split multiplicative | $4$ | \( 1369 = 37^{2} \) |
| $7$ | split multiplicative | $8$ | \( 8214 = 2 \cdot 3 \cdot 37^{2} \) |
| $37$ | additive | $722$ | \( 42 = 2 \cdot 3 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3.
Its isogeny class 57498k
consists of 3 curves linked by isogenies of
degrees dividing 9.
Twists
The minimal quadratic twist of this elliptic curve is 1554n2, its twist by $37$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{37}) \) | \(\Z/3\Z\) | not in database |
| $2$ | \(\Q(\sqrt{-111}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.6216.1 | \(\Z/2\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-3}, \sqrt{37})\) | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $6$ | 6.0.240177885696.1 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.2.47856143952.2 | \(\Z/9\Z\) | not in database |
| $6$ | 6.2.1429630272.1 | \(\Z/6\Z\) | not in database |
| $6$ | 6.0.4288890816.1 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $18$ | 18.0.2542727022271664105435141294247954591.3 | \(\Z/9\Z\) | not in database |
| $18$ | 18.2.1901357556235080161602346854737567811633152.1 | \(\Z/18\Z\) | 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 | nonsplit | split | ord | split | ord | ord | ss | ord | ord | ord | ord | add | ord | ord | ord |
| $\lambda$-invariant(s) | 3 | 4 | 1 | 2 | 1 | 1 | 1,1 | 1 | 1 | 1 | 1 | - | 1 | 1 | 1 |
| $\mu$-invariant(s) | 0 | 1 | 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.