Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy=x^3-x^2-33117x-1667459\)
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(homogenize, simplify) |
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\(y^2z+xyz=x^3-x^2z-33117xz^2-1667459z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-529875x-107247250\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(-57, 209)$ | $1.1144118510161890415587530749$ | $\infty$ |
| $(855/4, 7087/8)$ | $3.2026675071107928450179244721$ | $\infty$ |
| $(-589/4, 589/8)$ | $0$ | $2$ |
Integral points
\( \left(-147, 163\right) \), \( \left(-147, -16\right) \), \( \left(-57, 209\right) \), \( \left(-57, -152\right) \), \( \left(209, 608\right) \), \( \left(209, -817\right) \), \( \left(273, 2959\right) \), \( \left(273, -3232\right) \), \( \left(3059, 167333\right) \), \( \left(3059, -170392\right) \), \( \left(4275, 277096\right) \), \( \left(4275, -281371\right) \), \( \left(128459, 45976808\right) \), \( \left(128459, -46105267\right) \)
Invariants
| Conductor: | $N$ | = | \( 59850 \) | = | $2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \cdot 19$ |
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| Discriminant: | $\Delta$ | = | $1111458938625000$ | = | $2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 7 \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( \frac{9521387989875}{2634569336} \) | = | $2^{-3} \cdot 3^{9} \cdot 5^{3} \cdot 7^{-1} \cdot 19^{-6} \cdot 157^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $1.5944480594337099843840851782$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $0.51507603104963237423489420236$ |
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| $abc$ quality: | $Q$ | ≈ | $1.0103258818153782$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $3.8944178789013786$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 2$ |
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| Mordell-Weil rank: | $r$ | = | $ 2$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $3.3791432803653113905897363916$ |
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| Real period: | $\Omega$ | ≈ | $0.36128706289248801298691281081$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 24 $ = $ 1\cdot2\cdot2\cdot1\cdot( 2 \cdot 3 ) $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $2$ |
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| Special value: | $ L^{(2)}(E,1)/2!$ | ≈ | $7.3250445051364230654756929879 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 7.325044505 \approx L^{(2)}(E,1)/2! & \overset{?}{=} \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.361287 \cdot 3.379143 \cdot 24}{2^2} \\ & \approx 7.325044505\end{aligned}$$
Modular invariants
Modular form 59850.2.a.t
For more coefficients, see the Downloads section to the right.
| Modular degree: | 331776 |
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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 5 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 |
| $3$ | $2$ | $III$ | additive | 1 | 2 | 3 | 0 |
| $5$ | $2$ | $I_0^{*}$ | additive | 1 | 2 | 6 | 0 |
| $7$ | $1$ | $I_{1}$ | nonsplit multiplicative | 1 | 1 | 1 | 1 |
| $19$ | $6$ | $I_{6}$ | split multiplicative | -1 | 1 | 6 | 6 |
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 |
|---|---|---|
| $2$ | 2B | 2.3.0.1 |
| $3$ | 3B | 3.4.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 15960 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 19 \), index $96$, genus $1$, and generators
$\left(\begin{array}{rr} 1 & 6 \\ 6 & 37 \end{array}\right),\left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 9586 & 3195 \\ 11145 & 6376 \end{array}\right),\left(\begin{array}{rr} 15949 & 12 \\ 15948 & 13 \end{array}\right),\left(\begin{array}{rr} 10251 & 6520 \\ 3830 & 9701 \end{array}\right),\left(\begin{array}{rr} 2746 & 3195 \\ 885 & 6376 \end{array}\right),\left(\begin{array}{rr} 9056 & 6395 \\ 10605 & 12736 \end{array}\right),\left(\begin{array}{rr} 11 & 2 \\ 15910 & 15951 \end{array}\right),\left(\begin{array}{rr} 4201 & 12780 \\ 15630 & 12841 \end{array}\right),\left(\begin{array}{rr} 6383 & 0 \\ 0 & 15959 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 12 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[15960])$ is a degree-$91500104908800$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/15960\Z)$.
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 | $4$ | \( 525 = 3 \cdot 5^{2} \cdot 7 \) |
| $3$ | additive | $6$ | \( 175 = 5^{2} \cdot 7 \) |
| $5$ | additive | $14$ | \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \) |
| $7$ | nonsplit multiplicative | $8$ | \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \) |
| $19$ | split multiplicative | $20$ | \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2, 3 and 6.
Its isogeny class 59850f
consists of 4 curves linked by isogenies of
degrees dividing 6.
Twists
The minimal quadratic twist of this elliptic curve is 2394a4, its twist by $-15$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{42}) \) | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $2$ | \(\Q(\sqrt{5}) \) | \(\Z/6\Z\) | not in database |
| $4$ | 4.0.54583200.1 | \(\Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{5}, \sqrt{42})\) | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $6$ | 6.0.656373375.5 | \(\Z/6\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/12\Z\) | 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 |
| $16$ | deg 16 | \(\Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/12\Z\) | not in database |
| $18$ | 18.6.21016246813227928820669132102637277512000000000.2 | \(\Z/18\Z\) | 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 | add | add | nonsplit | ss | ord | ord | split | ord | ord | ord | ord | ord | ord | ss |
| $\lambda$-invariant(s) | 9 | - | - | 2 | 2,2 | 2 | 4 | 3 | 2 | 2 | 2 | 2 | 2 | 2 | 2,2 |
| $\mu$-invariant(s) | 1 | - | - | 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.