Minimal Weierstrass equation
Minimal Weierstrass equation
Simplified equation
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\(y^2+xy+y=x^3-275327368x+1652782131158\)
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(homogenize, simplify) |
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\(y^2z+xyz+yz^2=x^3-275327368xz^2+1652782131158z^3\)
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(dehomogenize, simplify) |
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\(y^2=x^3-356824268307x+77113273584124206\)
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(homogenize, minimize) |
Mordell-Weil group structure
\(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
Mordell-Weil generators
| $P$ | $\hat{h}(P)$ | Order |
|---|---|---|
| $(95044, 28832465)$ | $4.8208488578458891829009228218$ | $\infty$ |
| $(-19031, 9515)$ | $0$ | $2$ |
| $(7593, -3797)$ | $0$ | $2$ |
Integral points
\( \left(-19031, 9515\right) \), \( \left(7593, -3797\right) \), \( \left(95044, 28832465\right) \), \( \left(95044, -28927510\right) \)
Invariants
| Conductor: | $N$ | = | \( 35490 \) | = | $2 \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2}$ |
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| Discriminant: | $\Delta$ | = | $155636722957847040000000000$ | = | $2^{18} \cdot 3^{2} \cdot 5^{10} \cdot 7^{2} \cdot 13^{10} $ |
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| j-invariant: | $j$ | = | \( \frac{478202393398338853167169}{32244226560000000000} \) | = | $2^{-18} \cdot 3^{-2} \cdot 5^{-10} \cdot 7^{-2} \cdot 13^{-4} \cdot 271^{3} \cdot 288559^{3}$ |
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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: | $h_{\mathrm{Faltings}}$ | ≈ | $3.7746249531774171964927508688$ |
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| Stable Faltings height: | $h_{\mathrm{stable}}$ | ≈ | $2.4921502744466488284660071480$ |
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| $abc$ quality: | $Q$ | ≈ | $1.023218976193033$ | |||
| Szpiro ratio: | $\sigma_{m}$ | ≈ | $6.673091102409363$ | |||
BSD invariants
| Analytic rank: | $r_{\mathrm{an}}$ | = | $ 1$ |
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| Mordell-Weil rank: | $r$ | = | $ 1$ |
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| Regulator: | $\mathrm{Reg}(E/\Q)$ | ≈ | $4.8208488578458891829009228218$ |
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| Real period: | $\Omega$ | ≈ | $0.056583262550068365556050969882$ |
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| Tamagawa product: | $\prod_{p}c_p$ | = | $ 320 $ = $ 2\cdot2\cdot( 2 \cdot 5 )\cdot2\cdot2^{2} $ |
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| Torsion order: | $\#E(\Q)_{\mathrm{tor}}$ | = | $4$ |
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| Special value: | $ L'(E,1)$ | ≈ | $5.4555871327538231017418440788 $ |
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| Analytic order of Ш: | Ш${}_{\mathrm{an}}$ | ≈ | $1$ (rounded) |
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BSD formula
$$\begin{aligned} 5.455587133 \approx L'(E,1) & = \frac{\# ะจ(E/\Q)\cdot \Omega_E \cdot \mathrm{Reg}(E/\Q) \cdot \prod_p c_p}{\#E(\Q)_{\rm tor}^2} \\ & \approx \frac{1 \cdot 0.056583 \cdot 4.820849 \cdot 320}{4^2} \\ & \approx 5.455587133\end{aligned}$$
Modular invariants
For more coefficients, see the Downloads section to the right.
| Modular degree: | 15482880 |
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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 | $\mathrm{ord}_p(N)$ | $\mathrm{ord}_p(\Delta)$ | $\mathrm{ord}_p(\mathrm{den}(j))$ |
|---|---|---|---|---|---|---|---|
| $2$ | $2$ | $I_{18}$ | nonsplit multiplicative | 1 | 1 | 18 | 18 |
| $3$ | $2$ | $I_{2}$ | split multiplicative | -1 | 1 | 2 | 2 |
| $5$ | $10$ | $I_{10}$ | split multiplicative | -1 | 1 | 10 | 10 |
| $7$ | $2$ | $I_{2}$ | nonsplit multiplicative | 1 | 1 | 2 | 2 |
| $13$ | $4$ | $I_{4}^{*}$ | additive | 1 | 2 | 10 | 4 |
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 |
|---|---|---|
| $2$ | 2Cs | 2.6.0.1 |
The image $H:=\rho_E(\Gal(\overline{\Q}/\Q))$ of the adelic Galois representation has level \( 10920 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \), index $48$, genus $0$, and generators
$\left(\begin{array}{rr} 6891 & 2522 \\ 1846 & 8399 \end{array}\right),\left(\begin{array}{rr} 2523 & 2522 \\ 1118 & 8399 \end{array}\right),\left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 5461 & 5044 \\ 2522 & 10089 \end{array}\right),\left(\begin{array}{rr} 5253 & 5044 \\ 7982 & 2523 \end{array}\right),\left(\begin{array}{rr} 6719 & 0 \\ 0 & 10919 \end{array}\right),\left(\begin{array}{rr} 1 & 0 \\ 4 & 1 \end{array}\right),\left(\begin{array}{rr} 7801 & 2522 \\ 0 & 1 \end{array}\right),\left(\begin{array}{rr} 10917 & 4 \\ 10916 & 5 \end{array}\right)$.
The torsion field $K:=\Q(E[10920])$ is a degree-$38954430627840$ Galois extension of $\Q$ with $\Gal(K/\Q)$ isomorphic to the projection of $H$ to $\GL_2(\Z/10920\Z)$.
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 | $4$ | \( 169 = 13^{2} \) |
| $3$ | split multiplicative | $4$ | \( 5915 = 5 \cdot 7 \cdot 13^{2} \) |
| $5$ | split multiplicative | $6$ | \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \) |
| $7$ | nonsplit multiplicative | $8$ | \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \) |
| $13$ | additive | $98$ | \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \) |
Isogenies
This curve has non-trivial cyclic isogenies of degree $d$ for $d=$
2.
Its isogeny class 35490bk
consists of 4 curves linked by isogenies of
degrees dividing 4.
Twists
The minimal quadratic twist of this elliptic curve is 2730z2, its twist by $13$.
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 $E(\Q)_{\rm tors}$ $\cong \Z/{2}\Z \oplus \Z/{2}\Z$ are as follows:
| $[K:\Q]$ | $K$ | $E(K)_{\rm tors}$ | Base change curve |
|---|---|---|---|
| $4$ | \(\Q(\sqrt{91}, \sqrt{105})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-14}, \sqrt{26})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $4$ | \(\Q(\sqrt{-26}, \sqrt{30})\) | \(\Z/2\Z \oplus \Z/4\Z\) | not in database |
| $8$ | deg 8 | \(\Z/2\Z \oplus \Z/6\Z\) | not in database |
| $16$ | deg 16 | \(\Z/4\Z \oplus \Z/4\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | not in database |
| $16$ | deg 16 | \(\Z/2\Z \oplus \Z/8\Z\) | 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 | split | nonsplit | ss | add | ord | ord | ord | ord | ord | ord | ord | ss | ord |
| $\lambda$-invariant(s) | 4 | 2 | 2 | 1 | 1,1 | - | 1 | 3 | 3 | 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 | 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.