Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-x^2-307640x-133337285\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-x^2z-307640xz^2-133337285z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-4922235x-8538508458\)
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(homogenize, minimize) |
Mordell-Weil group structure
trivial
Invariants
| Conductor: | $N$ | = | \( 58482 \) | = | $2 \cdot 3^{4} \cdot 19^{2}$ |
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| Discriminant: | $\Delta$ | = | $-5825913898231922688$ | = | $-1 \cdot 2^{21} \cdot 3^{10} \cdot 19^{6} $ |
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| j-invariant: | $j$ | = | \( -\frac{1159088625}{2097152} \) | = | $-1 \cdot 2^{-21} \cdot 3^{2} \cdot 5^{3} \cdot 101^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $2.2908937127949152575329765851$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $-0.096836017345063048634241494948$ |
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| $abc$ quality: | $Q$ | ≈ | $1.1123490200903752$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $4.640933177648703$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 0$ |
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| Mordell-Weil rank: | $r$ | = | $ 0$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | = | $1$ |
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| Real period: | $\Omega$ | ≈ | $0.095591640465082621596751608968$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 63 $ = $ ( 3 \cdot 7 )\cdot3\cdot1 $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $1$ |
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| Special value: | $ L(E,1)$ | ≈ | $6.0222733493002051605953513650 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | = | $1$ (exact) |
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BSD formula
$$\begin{aligned} 6.022273349 \approx L(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.095592 \cdot 1.000000 \cdot 63}{1^2} \\ & \approx 6.022273349\end{aligned}$$
Modular invariants
Modular form 58482.2.a.v
For more coefficients, see the Downloads section to the right.
| Modular degree: | 904932 |
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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 3 primes $p$ of bad reduction:
| $p$ | Tamagawa number | Kodaira symbol | Reduction type | Root number | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $21$ | $I_{21}$ | split multiplicative | -1 | 1 | 21 | 21 |
| $3$ | $3$ | $IV^{*}$ | additive | -1 | 4 | 10 | 0 |
| $19$ | $1$ | $I_0^{*}$ | additive | -1 | 2 | 6 | 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 | 8.2.0.1 |
| $3$ | 3B | 3.4.0.1 |
| $7$ | 7B | 7.8.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \), index $768$, genus $21$, and generators
$\left(\begin{array}{rr} 7582 & 2337 \\ 1197 & 3193 \end{array}\right),\left(\begin{array}{rr} 4105 & 6954 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5321 & 7448 \\ 2128 & 7449 \end{array}\right),\left(\begin{array}{rr} 1 & 3192 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 7980 & 1 \end{array}\right),\left(\begin{array}{rr} 8567 & 0 \\ 0 & 9575 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 3192 & 1 \end{array}\right),\left(\begin{array}{rr} 3991 & 3306 \\ 7980 & 799 \end{array}\right),\left(\begin{array}{rr} 1 & 1938 \\ 5586 & 4789 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 1008 & 1 \end{array}\right),\left(\begin{array}{rr} 759 & 7334 \\ 1064 & 1671 \end{array}\right),\left(\begin{array}{rr} 1597 & 5586 \\ 1197 & 5587 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 3990 & 1 \end{array}\right),\left(\begin{array}{rr} 8569 & 1008 \\ 8568 & 8569 \end{array}\right),\left(\begin{array}{rr} 6385 & 3192 \\ 6384 & 3193 \end{array}\right),\left(\begin{array}{rr} 799 & 5586 \\ 6783 & 8779 \end{array}\right),\left(\begin{array}{rr} 1 & 2736 \\ 0 & 1 \end{array}\right)$.
The torsion field $K:=\Q(E[9576])$ is a degree-$1930080337920$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/9576\Z)$.
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 |
|---|---|---|---|
| $2$ | split multiplicative | $4$ | \( 29241 = 3^{4} \cdot 19^{2} \) |
| $3$ | additive | $4$ | \( 19 \) |
| $7$ | good | $2$ | \( 29241 = 3^{4} \cdot 19^{2} \) |
| $19$ | additive | $182$ | \( 162 = 2 \cdot 3^{4} \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
3, 7 and 21.
Its isogeny class 58482z
consists of 4 curves linked by isogenies of
degrees dividing 21.
Twists
The minimal quadratic twist of this elliptic curve is 162c4, its twist by $-19$.
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 |
|---|---|---|---|
| $2$ | \(\Q(\sqrt{57}) \) | \(\Z/3\Z\) | not in database |
| $3$ | 3.1.648.1 | \(\Z/2\Z\) | not in database |
| $6$ | 6.0.3359232.4 | \(\Z/2\Z \oplus \Z/2\Z\) | not in database |
| $6$ | 6.0.405017091.1 | \(\Z/3\Z\) | not in database |
| $6$ | 6.0.2269040749479.8 | \(\Z/7\Z\) | not in database |
| $6$ | 6.2.8640364608.3 | \(\Z/6\Z\) | not in database |
| $12$ | deg 12 | \(\Z/4\Z\) | not in database |
| $12$ | deg 12 | \(\Z/3\Z \oplus \Z/3\Z\) | not in database |
| $12$ | deg 12 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $12$ | 12.0.5148545922796222038771441.3 | \(\Z/21\Z\) | not in database |
| $18$ | 18.6.5356378900275677237648054726627328.2 | \(\Z/9\Z\) | not in database |
| $18$ | 18.0.114269416539214447736491834168049664.1 | \(\Z/6\Z\) | not in database |
| $18$ | 18.0.3062434496351722360338814830646612260028416.1 | \(\Z/14\Z\) | 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 | ss | ord | ord | ord | ord | add | ord | ord | ord | ord | ord | ord | ord |
| $\lambda$-invariant(s) | 4 | - | 0,0 | 0 | 0 | 0 | 0 | - | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| $\mu$-invariant(s) | 0 | - | 0,0 | 1 | 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
All $p$-adic regulators are identically $1$ since the rank is $0$.